AKADEMIE DER WISSENSCHAFTEN DER DDR
ZENTRALINSTITUT
FUR
MATHEMATIK UND MECHANIK
REPORT
Reinhard Pijschel
A general Galois theory
for operations and relations
and concrete characterization
of related algebraic structures
( C o m m u n i c a t e d by H. K o c h )
B e r l i n 1980
Keywords
General Galois theory
Concrete characterization
Clone of operations
Clone of relations
Polymorphism
Invariant relation
AMS Subject classification (1980)
08-00, 08A30, 08435, 08A40, 064159 2°B10
Received January loth, 1980
ABSTRACT
The property of an operation to preserve a relation induces
a Galois connection between sets of operations and relations,
resp. This Galois connection (Pol-Inv), for operations and
relations on an arbitrary set will be investigated in the
present papercpart I). The Galois closed sets can be characterized as local closures of clones of operations or relations, resp. These results are applied to concrete characterization problems (part 2). In particular, the concrete characterization of automorphism groups, endomorphism semigroups,
subalgebra lattices and congruence lattices of universal (or
relational) algebras is treated in detailed form.
ZUSATDdIENPASSUNG
Die Eigenechaft einer Funktion, eine Relation zu bewahren
induziert eine Galoieverbindung zwischen Punktionen- und
Relationenmengen. In der vorliegenden Arbeit wird diese Galoisverbindung (.Pol-Inv) fiir Operationen und Relationen auf
einer beliebigen Menge untersucht (Teil 1). Die Galois-abgeschlosaenen Mengen werden als lokale AbschlieBungen von
Operationen- bzw. Relationenklons characterisiert. Diese
Ergebnisse werden auf konkrete CliarakterisierungsprobLeme
angewendet (Teil 2). Dabei wird besonders auf die konkrete
Charakterisierung von Automorphismengruppe, Endomorphismenhalbgruppe, Unteralgebrenverband und Kongruenzrelationenverband universaler (2.T. auch relationaler) Algebren eingegangen.
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Introduction
........................................
5
Part 1
Clones and the Galois connection Pol-Inv
P.
...................... 11
...............................
20
................................
23
Definitions and Preliminaries
52 Clones of operations
53 Clones of relations
$ 4 The Galois connection Pol-Inv
$1
.....................
31
Part
2
.
.
Concrete characterization of related algebraic structures
.
(Characterization of operational systems via
relational ones) ...................................37.
$5 Concrete characterization I
6
.
Concrete characterization I1
(Characterization of relational systems via
universal algebras)
$7 Concrete characterization I11
(Specialized problems)
................................ 43
.
............................. 48
$8 Concrete characterization of Aut 01 ................. 51
$9 Concrete characterization of ~ n d u................. 58
$10 Concrete characterization of Subi? ................. 66
$11 Concrete characterization of Con CZ ................. 69
$12 Concrete characterization of Aut t q Sc Sub i'l ......... 71
$13 Concrete cl~aracte~ization
of Rut u[ & Cona ......... 78
$14 Concrete characterization of End e1 & Sub@ ......... 82
.
$1 5 Conore t e characterizat ione IV
(Survey on related Galois aonneotions)
$16 Krasner-clones of relations
REPERmCES
............ 84
....................... 87
............................................
SUBJECT INDEX............o..
INDEX 0% NOTATIONS
92
100
....................................
I01
Every u n i v e r e a l a l g e b r a a =(A;F> i s a s s o c i a t e d w i t h soc a l l e d r e l a t e d s t r u c t u r e s , e.g., w i t h t h e l a t t i c e s
~ u b aand Con& of i t s s u b a l g e b r a s and congruences respect i v e l y o r w i t h i t s group A u t a of automorphisms.
Such r e l a t e d s t r u c t u r e s have a common p r o p e r t y : f o r i n s t a n c e , B t S u b a , BeConCf and ~ G A aU ~ can be c o n s i d e r e d as
r e l a t 5 0 n s of a s p e c i a l k i n d ( h e r e unary o r equivalence r e l a t i o n s and permutations f : A -+A
c o n s i d e r e d as s u b s e t s o f
A x A ) which a r e p r e s e r v e d by a l l (fundamental) o p e r a t i o n s
Therefore, i n g e n e r a l , w e a r e i n t e r e s t e d i n
g t F of fl ah;@.
r e l a t i o n s on A which a r e i n v a r i a n t f o r ( i . e . preserved
by) a l l grF.
The p r o p e r t y ' t o p e r a t i o n p r e s e r v e s r e l a t i o n " i n d u c e s a
G a l o i s c o n n e c t i o n between s e t s of o p e r a t i o n s and r e l a t i o n s ,
r e s p . F o r a s e t Q (P, r e s p . ) of r e l a t i o n s ( o p e r a t i o n s ) , l e t
P o l Q ( I n v F ) be t h e s e t of a l l o p e r a t i o n s ( r e l a t i o n s ) which
preserve (are invariant f o r ) a l l r e l a t i o n s i n Q (operations
i n I?, r e s p e c t i v e l y ) ! . Then t h e o p e r a t o r s P o l - I n v e s t a b l i s h
t h e mentioned G a l o i s connection.
That what we c a l l "General G a l o i s t h e o r y f o r o p e r a t i o n s
and r e l a t i o n s n i n c l u d e s mainly t h e f o l l o w i n g t o p i c s :
a ) I n v e s t i g a t i o n of t h e G a l o i s connection P o l - I n v (and
s e v e r a l m o d i f i c a t i o n s and r e s t r i c t i o n s of i t ) ;
b ) C h a r a c t e r i z a t i o n of t h e G a l o i s c l o s e d s e t s ;
c ) I n v e s t i g a t i o n of p r o p e r t i e s of o p e r a t i o n a l systems by
means o f p r o p e r t i e s of r e l a t e d systems of i n v a r i a n t
r e l a t i o n s (and v i c e v e r s a ) .
It t u r n s o u t t h a t there a r e c l o s e c o n n e c t i o n s ( i n p a r t i c u l a r of b ) ) t o s o - c a l l e d
concrete c h a r a c t e r i z a t i o n
t h e c h a r a c t e r i z a t i o n of t h e
p r o b l e m s , t h a t means, e.g.,
l a t t i c e s SubB , con& and o f t h e group Aut C?
f o r a l l alg e b r a s w i t h u n d e r l y i n g s e t A as s e t s o f s u b s e t s o f A , s e t a
o f p a r t i t i o n s on A and s e t s of permutations on A , r e s p .
In fact, a s e t
of r e l a t i o n s i s t h e s e t of a l l i n v a r i a n t r e l a t i o n s of a universal algebra
Q=
I n v P o l Q. The o n l y problem now c o n s i s t s i n f i n d i n g s u i t a b l e
c l o s u r e o p e r a t i o n s t o d e f i n e I k l o n e s o f r e l a t i o n s " [Q] i n
s u c h a way t h a t t h e y c o i n c i d e w i t h t h e G a l o i s c l o s u r e
I n v P o l Q.
Note t h a t
by t h e above o b s e r v a t i o n s ( f o r more d e t a i l s
s e e $1 5 )
every s o l u t i o n of a concrete c h a r a c t e r i z a t i o n
problem p r o v i d e s a c h a r a c t e r i z a t i o n o f a G a l o i s c l o s u r e and
v i c e versa.
Q
iff
-
-
I n t h e p r e s e n t p a p e r we a r e m a i n l y c o n c e r n e d w i t h r e s u l t s
on t h e G a l o i s c o n n e c t i o n P o l I n v ( P a r t 1' ) and t h e a p p l i c a t i o n o f t h e s e r e s u l t s t o c o n c r e t e c h a r a c t e r i z a t i o n problems ( P a . r t 2 ) . F i r s t o f a11 w e w i l l l i s t some r e f e r e n c e s
which a l s o r e f l e c t t h e h i s t o r i c a l developement o f o u r t o p i c .
-
-------
---
-------- ---
e l a t i o n s : +)
General G
alois theory f o r o
------------ ~ e r a t i o n sand r---------
...
KRASNER(1,938, ,1976) ( m a i n l y f o r p e r m u t a t i o n g r o u p s and
t r a n s f o r m a t i o n s e m i g r o u p s ) ; IWZ~,~ECOV(~
959 ) ; GEIGER(I9 6 8 ) ;
B~DNAREUK/KALUZWIN/K~TOV/RO~IOV
(1 969 ) ; POSCHEL/KALUZNIN
(197s)) ( f o r f i n i t e s e t s A); ROSEKJ3ERG(l972,... , 1 9 7 9 ) ;
POIZAT (1 971 , ,1 979 ) , KRASI~XR/POI~,AT(~
976) (most g e n e r a l
c a s e ) ; P0s~HEL(1973)( f o r h e t e r o g e n e o u s a l g e b r a s ) ; LECO?,ITE
(1 976/77) ; SAUER/STOBE(1977/78); ROE.IOV(1977); POSCIIEL(1979 )i
FLEI SCHER ( 1978 )
A b s t r a c t c h a r a c t e r i z a t i o n e r------oblems ( i . e . c h a r a c t e r i z a t i o n o f
:
r e l a t e d s t r u c t u r e s "up t o
...
.
-------- ----------------
BIHECHOFF( I 9 4 6 ) ; BIRKHOFF/PRINK(I 948 ) ; GRATZER/SCI~~"II~DT
(1 9 6 3 ) ; E.T. sCIDIIDT(~963/64); G R A T Z F ; R / L ~ ~ I9P6E7()~;
J ~ N S S O N(1974) ; SCIIEIN/TROHII;I;IEP~KO(1 979 1.
C o n c r e t e ---------------characterization
--------
~roblems
ICRASMER (1 950 ) ; ARPI1J.3~U~T/~~~TiVI1~~
( I 970 ) ;
Automorphism group :
~ 6 1 ? ~ ~ 0 ~ ( 1 9 PWNKA(1968);
68);
J ~ N s s o I J ( ~ ~GOULD(~
~ ~ ) ; 972a)
( f o r a l g e b r a s o f f i n i t e t y p e ) ; SZJ& (1 9 7 5 ) , BREDIHIN(1976)
( f o r l o c a l automorphiams)
')
c f . a l s o 94, p. 3 2 .
.
Endomorphism semigroup (Problem 3 i n [Gr])
:
LAMPE(1968);
GR~TZER/LAI~PE(~
9 6 8 ) ; J E ~ E K972);
(~
STOI\IE(1969/75);
~ ~ A I 3 6 ( 1 9 7 8 ) (; c f . [ ~ b n 7 u ) .
BIRKHO~'F/FRI~IK(~
948) ( c f .fibn72]) ;
Subalgebra l a t t i c e :
JoBNsoN/sEIFERT (1967 ) ( f o r unary a l g e b r a s ) ; GOULD(1968 )
( i n p a r t i c u l a r f o r a l g e b r a s of f i n i t e t y p e ) .
AFUIIBRUST(~970 ) ;
S.BURRS/II.CRAPO/A.DAY/D.HIGGS/W.NICKOLS(C~ . [ J b n 7 2 ( ~ . 1 ' 7 4 n
QUACKENBUSM/~VOLK (1 971 ) ; J ~ N S S O N972
( ~ (Thm. 4 4 1 ) ) ; D R A ~ K O VIE OVA(^ 974) ; \VI3RI'JER(1#974).
Congruence l a t t i c e (Problem 2 i n [~r]) :
.
STONE(1972);
Automorphism group & subalgebra l a t t i c e :
~OULD(197,2b)(fora l g e b r a s of f i n i t e t y p e ) .
Endomorphism seniigroup & s u b a l g e b r a l a t t i c e :
SAUER/STOIG(I
977a); ( c f .[Jbn74-).
Automorphism group & congruence l a t t i c e :
W E R N E R (974)
~
(Problem 4 and c o n j e c t u r e ~ v ~ N ( P 452.)]).
.
All
structures t o g e t h e r (automorphisms, endomorphisms, sub-
a l g e b r a s , congruences) :
S Z A B ~ (978);
I
POSCHEL(1979).
General systems o f r e l a t i o n s ( i n p a r t i c u l a r s u b a l g e b r a l a t t i c e of c a r t e s i a n powers o f a l s e b r a s ) :
B~DIJAR~UK/KALU~NIN/KO!POV/ROMOV
(1 9 69 ) ; DANTONI (1 9 69 ) ;
POSCHEL/KALUZNITI (1 979 ) ; S Z A B(1~978 ) ; PUSCIIEL(I979 ) ;
ROSEBDERG (1 978 )
.
E x p l i c i t e l y we want t o mention t h e work of hl. Krasner,
D. Geiger and L. Szabb. Id. Krasner w a s t h e f i r s t who s y s t e m a t i c a l l y i n v e s t i g a t e d G a l o i s c o n n e c t i o n s between (unary)
o p e r a t i o n s and r e l a t i o n s ( t h e r e f o r e , i n $16, we i n t r o d u c e
t h e n o t i o n I1Krasner-clonet1). D. Geiger o b t a i n e d t h e main r e s u l t s of [ ~ o / ~ a l ] , t o o , and o u t l i n e d how t o i n v e s t i g a t e t h e
i n f i n i t e case. L. Szab6 c h a r a c t e r i z e d i n [Sz38] c l o n e e of
r e l a t i o n s by a c l o s u r e p r o p e r t y w i t h r e s p e c t t o s o - c a l l e d
formula schemes and ( i n d e p e n d e n t l y ) o b t ~ i n e d .e s s e n t i a l l y t h e
same r e s u l t s concerning some of t h e c o n c r e t e c h a r a c t e r i z a t i o n
problems.
We s h o r t l y o u t l i n e now t h e c o n t e n t o f t h e p r e s e n t p n p e r .
In
we i n t r o d u c e o r r e c a l l n e a r l y a l l used no-Lions and
n o t a t i o n s and g i v e some p r e l i m i n a r y r e s u l t s .
In
c l o n e s of o p e r a t i o n s a r e c o n s i d e r e d and some prop e r t i e s w i t h r e s p e c t t o i n v a r i a n t r e l a t i o n s a r e given.
In
c l o n e s of r e l a t i o n s a r e i n t r o d u c e d (and m o t i v a t e d ) .
Some c l o s u r e p r o p e r t i e s o f t h e s e c l o n e s a r e g i v e n .
I n $q t h e G a l o i s c l o s e d s e t s o f o p e r a t i o n s and r e l a t i o n s
( w i t h r e s p e c t t o P o l - I n v ) a r e c h a r a c t e r i z e d as l o c a l c l o s u r e s o f c l o n e s o f o p e r a t i o n 3 and r e l a t i o n s , r e s p e c t i v e l y .
R e s t r i c t i o n s on t h e a r i t y o f o p e r a t i o n s and r e l a t i o n s under
consideration a r e i n v e s t i g a t e d , too.
In
w e c o n s i d e r t h e c o n c r e t e c h a r a c t e r i z a t i o n problem
how t o c h a r a c t e r i z e t h o s e s e t s F o f o p e r a t i o n s which a r e
I t r e l a t e d t t t o some r e l a t i o n a l system % =<A; related lneans
e.g. F = ~ u f ; & ,F = H O ~ ( & ~ , & , ) F
, = P o l Q, etc.).
In
t h e f o l l o w i n g c o n c r e t e c h a r a c t e r i z a t i o n problem i s
c o n s i d e r e d : How t o c h a r a c t e r i z e t h o s e s e t s Q o f r e l a t i o n s
which a r e " r e l a t e d t t t o some u n i v e r s a l a l g e b r a fl
t e d means e . g . Q = c o n f l , ~ = h ~ t B Qt =, S U ~d n , Q = I n v F).
There a r e a l s o some c o n t r i b u t i o n s t o t h e c a s e where & i s o f
bounded r a n k o r has a f i n i t e system of (fundamental) opexa%ions.
rela- la-
The problems c o n s i d e r e d i n $ 6 w i l l be s p e c i a l i z e d i n
where we answer t h e q u e s t i o n f o r a simultaneous c o n c r e t e char a c t e r i z a t i i o n of automorphism group, endomorphism monoid,
s u b a l g e b r a and congruence l a t t i c e s of u n i v e r s a l a l g e b r a s .
O f c o u r s e , t h e s e s p e c i a l r e l a t e d s t r u c t u r e s a r e of grea*
algebraic i n t e r e s t . Therefore t h e concrete c h a r a c t e r i z a t i o n
of ('one o r s i m u l t a n e o u s l y two o f ) t h e s e s t r u c t u r e s w i l l be
--"-..-"-
t r e a t e d i n d e t a i l e d form i n $98-1 4. A s u r v e y on many, p a r t i a l l y well-known, r e s u l t s i s g i v e n and t h e a p p l i c a t i o n o f
t h e G e n e r a l G a l o i s t h e o r y i s d e m o n s t r a t e d . T h i s y i e l d s somet i m e s new r e s u l t s , sometimes new p r o o f s f o r known r e s u l t s .
The r e s u l t s o f $6 p r o v i d e a n answer t o a l l c o n c r e t e c h a r a c t e r i z a t i o n problems o f r e l a t e d s t r u c t u r e s o f a l g e b r a s i n t e r m s
of c l o n e s o f r e l a t i o n s ( o r o p e r a t i o n s ) . . N e v e r t h e l e s s we t h i n k
t h e s e r e s u l t s n o t t o be f i n a l o n e s b e c a u s e i n s p e c i a l c a s e s
s i m p l e r c h a r a c t e r i z a t i o n s might be o b t a i n e d ( a n d c a n be obt : a i n e d a s shown sometimes i n $58-14). Thus, more o r l e s s exp l i c i t e l y , a l o t o f problems f o r f u r t h e r i n v e s t i g a t i o n s i s
c o n t a i n e d i n 998-1 4.
-
I n 915 we w i l l e x p l a i n e once more e x p l i c i t e l y t h e i n t e r dependences o f c o n c r e t e c h a r a c t e r i z a t i o n problems and t h e
c h a r a c t e r i z a t i o n of Galois closed s e t s with r e s p e c t t o a can a x k a l l y r e l a t e d G a l o i s c o n n e c t i o n . From t h i s p o i n t o f view
we summarize t h e r e s u l t s g i v e n i n p r e v i o u s paragraphe(S54-14)
by l i s t i n g which r e s u l t s c h a r a c t e r i z e p r o p e r l y which G a l o i s
connection.
-
n
we i n v e s t i g a t e i n v a r i a n t r e l a t i o n s o f u n a r y , i n
p a r t i c u l a r b i j e c t i v e o p e r a t i o n s . These r e l a t i o n s ( a r e c h a r a c t e r i z e d as G a l o i s c l o ~ e ds e t s o f t h e c o r r e s p o n d i n g G a l o i s
c o n n e c t i o n Inv End o r I n v A u t a n d ) form s o - c a l l e d Krasn e r - c l o n e s ( o f l s t o r 2nd k i n d ) . The i n c l u s i o n s I n v Aut Q 2
=
3 Inv End Q 2 I n v P o l Q s u g g e s t t h a t K r a s n e r - c l o n e s m i g h t be
c h a r a c t e r i z e d by c l o s u r e p r o p e r t i e s which a r e somewhat s t r o n ger than those f o r ordinary clones of relations. This w i l l
be c l a r i f i e d i n $1 6.
-
-
I n t h e p a p e r , a l l d e f i n i t i o n s , p r o p o s i t i o n s e t c . a r e numb e r e d c o n s e c u t i v e l y ; t h e f i r s t number which o c c u r s on a page
i s marked a l s o o n t h e t o p o f t h i s page. The end o f a p r o o f
( o r of a s t a t e m e n t w i t h e a s y p r o o f ) i s marked by I.Referenc e s a r e g i v e n i n b r a c k e t s , sometimes w i t h some f u r t h e r i n f o r m a t i o n s i n p a r e n t h e s i s , e.g. [ ~ b n 7 2 ( ~ h m .3.6.411.
The f o l l o w i n g p i c t u r e shows t h e interdependence of t h e
paragraphs:
The o b j e c t i v e of t h e paper p r e s e n t e d h e r e i s t h e i n t r o d u c t i o n of a General G a l o i s t h e o r y ( f o r o p e r a t i o n s and r e l a t i o n s )
ae a h e l p f u l background f o r c o n c r e t e c h a r a c t e r i z a t i o n s of r e l a t e d a l g e b r a i c s t r u c t u r e s . IIowever , t h e r e remains many t h i n g s
t o do; e,g., t h e i n v o l v e d n o t i o n of a c l o n e of r e l a t i o n s needs
much more d e t a i l e d i n v e s t i g a t i o n s i n o r d e r t o g e t c o n d i t i o a s
which a r e "easyI1 t o check.
Almost a l l paragraphs of t h e p r e s e n t paper had been w r i t t e n
up d u r i n g 1977, b u t f o r n e a r l y two y e a r s t h e a u t h o r u n f o r t u n a t e l y c o u l d n o t f i n d time enough t o w r i t e up t h e f i n a l v e r s i o n .
Some r e s u l t s were o u t l i n e d i n a l e c t u r e g i v e n a t t h e conferenc e on tlAllgemeine Algebratt i n Klagenfurt ( A u s t r i a , May 1978);
a s h o r t n o t e ( i n which t h e p r e s e n t paper w a s r e f e r e d t o as a
p r e p r i n t ) waB p u b l i s h e d i n t h e proceeding^ of t h i s c o n f e r e n c e
( c f . f~ij7.91).
ACKIVOVJLEDGEE~IENT. I want t o e x p r e s s my t h a n k s t o P r o f e s s o r
L.A. Kaluknin who d i r e c t e d my a t t e n t i o n t o t h e i n v e s t i g a t i o n
of a "General G a l o i s theoxytt and showed a c o n t i n u o u s i n t e r e s t
on my worlc.
Part 1
CLONES AND THE GALOIS CONNECTION Pol-Inv
$1
11.1
-
Definitions and Preliminaries
Let A be an arbitrary set (with lAla2) and let m,neIN,
where IN
4
1,2,3,. ..) denotes the set of all natural numbers
(without zero). Let
={f
OA
1 f : An
RP)= f g l
-
-m =[0,1 ,...,m-lf . With
..
*cAm)
A)
and
, resp.,
we denote the sets of all n-ary operationa and m-ary relations, resp., on the set A. Let
u
0p) and
- nrB
Universal
RA =
uIN
me
(m)
R~
a l g e b r a s < ~ ; ( f ~ ) or
~~~
re
>l a f i o n a l '
a 1 g e b r a s (i.e. relational systems [Gr(p.BV)
(A;
(P,),,,)
of some similiarity type are denoted shortly by (A;F>
or
<A; Q) where F = { ~ ~iJe ~ Jand Q={*~] ic~), respectively, because
we are not interested in the type of these algebras.
All considerations will be restricted to finitary (except
nullary) operations and relations. But it should be mentioned
that most of the given results can be generalized to infinitary operations and relations (for this purpose one has to
substitute B by a suitable chosen limit ordinal
dinal
-
number); we refer to flr/PoU,[Poi80]
- or car-
for this app-
roach. To avoid some technical modifications, nullary operations (constants) are not considered in universal algebras.
For o u r purposes n u l l a r y o p e r a t i o n s can be r e p l a c e d by unary
constant operations.
xeAm of t h e c a r t e s i a n power
The components of elements
Am o f A
a r e denoted by
x ( i ) ( i e ~ ) ,i . e . ,
- or
=(x(O) ,...,~ ( m - 1 ) ) (sometimes a l s o
x
x
fx
~ E o ! " ) , xeAn, we w r i t e
m
For rl , , r n € A ,
If
...
f
(rl
f (xo,
..., xn-1
,...,xm-1 1).
).
,...,rn 1
d e n o t e s t h e m-tupel
1.2
-
for
x=(xo
( f ( r l ( i ) , . . . , r n (i)))iemm
The most i m p o r t a n t n o t i o n f o r o u r i n v e s t i g a t i o n s i s
t h e f o l l o w i n g one:
v---------a r i a n t
(m)
A r e l a t i o n geRA
f o r an o p e r a t i o n
---s t a b 1e
for
---------
f is a p o l y m
----o r p -.----hism
froA
("I
of 9 , f
.
i s s a i d t o be
(or
i n ,---
f ~------------r e s e r v e sg ,
---------a d m i t s 9,
9 is
, ,
n ) belongs t o g whenever
rl,
rn '9. The empty r e l a t i o n @ i s p r e s e r v e d by every
operation fgOA.
f )
if
f( r
...,
Note t h a t f p r e s e r v e s 9 i f f 9 i s ( t h e base s e t o f ) a subalgeb r a o f <A; f>m o r , e q u i v a l e n t l y , i f f
(A;#
1 .f
-
f
i s homomorphism of
into <~;g).
Sometimes, f o r f ~ o p ) we
, consider the r e l a t i o n
f ' ={(x o,...,xn-l
,xn) e A n + '
(
i n s t e a d o f t h e o p e r a t i o n f . Thus, f o r
F'
f ( ~ ~ , . . . , x ~)=xn
-~
F 5 0 A , the lldottedll
=if'(f t ~ )
w i l l remind us t h a t we have t o handle t h e o p e r a t i o n l i k e r e -
l a t i o n s . It i s w e l l known t h a t gcOA p r e a e r v e s f * G O A *
iff
g and
1.4
-
f
commute.
For
F=OA and
Q S R A we u s e t h e f o l l o w i n g n o t a t i o n s :
F ( " ) : = F ~ o ~ ) ,Q ( m ) : = Q n R( mA)
(n,mem),
P o l Q = PolAQ: = \fcoA 1 f p r e s e r v e s a l l 9
. Q
( ' --~ o l----y m o -------r p h i s m so f Q ) ,
PolAF := PolAFe
9
I n v F = ~ n v ~ ~ : 1 =9 {i s~ i nr v~a r i~a n t f o r a l l f e F j
( --------------i n v a r i a n t s of F ) ,
:= ( P o l Q)("), 1nv(*)F := ( I n v F). (m)
( ---------e n d o m o r p-------his m s
EndAQ:=Pol;)Q
w-AutAQ :=
,
o f <A;Q>),
S A n P o l A Q( -----weak a u,
t,
o,
m,
or Eh-------ism
s3)9
AutAQ r={fcsAI. 1 f , f - ' ] ~ ~ o l ~ Q( ---------3a u t o m o r p------.-hisms),
where
SA d e n o t e s t h e f u l l
symmetric
p e r m u t a t i o n s ) on A. We w r i t e s h o r t l y
Polfg), 1nv{f)
P o l Q,>...
... (yrRA,
,
Pol9
g r o u p (of a l l
, Inv
f
f c - O A ) C l e a r l y we have:
~ ~ o l ( ~ + ~ ) ~ ~ P End
o l Q
( a~P o) l (1
~ -1~ ).Q .2.
2 W-Aut Q 2 Aut
Inv F 2
-
,... f o r
... 2 I ~(n+l
v ) F -2 I ~ v ( " ) F 2
-
*.
Q
,
2
- I n v (1 ) p
1.5 Remarks.
------- (i) P O ~ ( " ) Qi s t h e s e t o f a l l homomorphisms of
t h e r e l a t i o n a l a l g e b r a <A;
i n t o (A;
Q>"
Q).
( i i ) ~ n v ( i~s )t h ~e s e t o f a l l (base s e t s o f ) s u b a l g e b r a s of
{A; F ) ~ ; i n p a r t i c u l a r , 1nv" ) F i s t h e s e t Sub @ o f a l l ouba l g e b r a s o f fl =(A;F).
*) I n fS?879] t h e y a r e c a l l e d autornorphisms.
( i i i ) (Pol F)' = O i n I n v F.
(iv)
( ~ 0 1 ' " ) ~ ) ._c
- i n v ( n + l ) ~ ,n c - J .
1.6
-
We d e f i n e , f r S A ------s t r o n ~
- l- y p r e s e r v e s
{ f ( r . ) l r t g j = ~, e
if
(m
gcRA
------------
,
o f t h e r e l a t i o n a l system <A;?>
a) A U t A Q
f
if
i s an isomorphism(cf.&rj)
o n t o <A;?>
i s the set of a11
-
. Then we have:
fcSA which s t r o n g l y p r e s e r v e
g€Q5RA.
each
-------
(P r o o f : j f ( r ) ( r q j =c j
p t Q. D ). One e a s i l y s e e s t h a t
also preserves
-
s u b g r o u ~o f
J f-' ( r ) ~ 9
for all rcf,
AutAQ
f-'
&
E
J
g
SA.
b) For f i n i t e
-we have
A
w-Aut Q = Aut Q
( s i n c e few-Aut Q
7
c
.
c
c
implies
i.e.
f n = f e...ofew-Aut
Q
and
but
c) i n g e n e r a l , w-Aut Q
------ ---
properly contains
(1 ,
...,-1 ,0,1
f: x w x + l , y=BoRA
fW1
0 w - ~ u tp , i .e . w - ~ u tp 7 ~ u gt .)
( E x a m ~ l :e A = z = [
f e w - ~ u t p but
Aut Q.
d) Nevertheless,
,2,...],
we have
w-AutAQ = AutAQ a l s o f o r i n f i n i t e A
if for a l l
---
non-trivial
f o r some ~ G O ~ t9-i"
.
7 -
(The
roof
------
1 g 1 < No or
Q = g'
75' E Q
i s l e f t t o t h e r e a d e r as a n e a s y e x e r c i s e . ) I
1,7 P r o p o s i t i o n .
G a l o i s connection
of oA.
-
9c Q bolds
The p p e r a t o r s P o l and Inv d e f i n e -a
between the s u b s e t s of RA and t h e s u b s e t s
2 p a r t i c u l a r -we have
F$FICOA
Inv F
2 Inv
~1
Pol Q 3 Pol
Q5Q'5RA
1.8
-
a',
F s _ P o l I n v F,
Q E Inv Pol Q,
P o l Q = P o l Inv P o l Q ,
I n v F = I n v P o l I n v F,
For FSOA, Q 5 R A and
6,cAm(rnr~l)we d e f i n e
T F ( e ) i s ( t h e base s e t o f ) t h e s u b a l g e b r a o f ( A ; F ) ~ generat e d by
Q
and belongs t o
Inv P
(since
Inv F
under a r b i t r a r y i n t e r s e c t i o n s ) , c f . 1 . 5 ( i i ) .
r,(9)
=
9 --,9
u
TP(r)
for all
1.9
-
=9
<=7
Thus
g tTnv F.
.w
let < 620
Moreover,
is closed
5 OInv
For t h e i n v e s t i g a t i o n o f " l o c a l p r o p e r t i e s n o f opera-
t i o n s and r e l a t i o n s we d e f i n e t h e foll-owing
s u r e
o
e r a t o r s
------ q-----------
l o c a 1 ____-----c l o-
(stIN, F 5 0 A , QSRA):
T h i s i~t h e s e t o f a l l n-ary o p e r a . t i o n s ( n e 3 i ) w i t h t h e prop e r t y t h a t f o r every s u b s e t
of
B
F
ments t h e r e e x i s t s a member of
Lot I? :=
i.e.,
f belongs t o
n
every f i n i t e s u b s e t .
s ele-
t h a t agrees with f
on B.
,
.-,OC
s e IN
Loc F
w i t h a t moet
iff
f
a g r e e s w i t h some
grF
on
9 (ma3T)
T h i s i s t h e s e t o f a l l m-ary r e l a t i o n s
f o r every subset B of
e x i s t s a member o f
contained i n
9
I,oc
-----dm
elements t h e r e
9
on
B
and i s
.
Q :=
n
s-LoC
Q.
S G 31
( i i ) A relation
BS Am
g n B ~ 6 gf
s
t h a t agrees with
Q
( i ) (Loc F) (")=Loc F ( " ) ,
Remarks.
for all
9 w i t h a t most
such t h a t
belongs t o
(LOC Q ) (")=Loc Q ( n )( n e m ) .
s-LOC Q ( i f a n d ) o n l y i f
w i t h l ~ l i st h e r e e x i s t s a
. We have
(8~s-LOC Q
<+
@€Q
c+aQ w i t h
,
I n t h e n e x t p r o p o s i t i o n s we c o l l e c t some p r o p e r t i e s o f
these l o c a l operators.
1.10Proposition.
-
I&FSF150A,
Q5QtSRA, s,sl&W. Then:
( i ) 1-LOCF 2
. c . ~
EI-LOCFZ(S+I)-LOCF~...2 LOC F 2 F.
(ii)l-LOCQz
...z
s - L O C Q ~ ( S + ~ ) - L O C Q ~2 LOC Q 2 Q.
...
-
---
( i i i ) A l l o p e r a t o r s d e f i n e d i n 1.9 a r e c l o s u r e o p e r a t o r a ;
i n p a r t i c u l a r -we have:
a-LOC F s S-LOC F '
LOC LOC F = LOC F
,
,
S-LOC Q ~ s - L O C Q '
,
LOC LOC Q = LOC Q
,
8-LOC s'-LOC F = ~ " - L O CF , S-LOC 8'-LOC Q = s"-LOC Q ,
where a t l = min{s, s I )
,
Loc s-Loc F = s-Loc Loc F = s-Loc F,
LOC s-LOC Q = s-LOC LOC Q = s-LOC Q ,
(iv)
For f i n i t e A
Loc F = F
we have
and LOC
Q = Q
-f o r a l l FgOA,
QsRA.
(v)
s-LOC F'
The p r o o - f s
= P'
for
s22.
e a s i l y f o l l o w from t h e d e f i n i t i o n s .
1
1.1 1 P r o p o s i t i o n .
S-LOC(POQ
~ ) = P O Q~
for
for
(1 u... ~ R L * ) ,S C J .
QsRA
()'
LOC(1nv P) = I n v F
for
FgOA.
b
s-LOC(InvF)=InvF
for
B ~ o ~ ) ~ . . . v oSEB.
~ ~ ) ,
(a)
Loc(Po1 Q ) = P o l Q
(b)
Q &RA.
-- -
(c)
mention h e r e t h a t , i n g e n e r a l , Loc G
-
if
GgSA. But a l l
i s n o t con---
-
tained i n
SA
Proof.
F o r ( a ) - ( b t ) we have t o sliow t h a t t h e l e f t s i d e
----_oo
faLoc SA a r e i n j e c t i v e .
i s c o n t a i n e d i n t h e r i g h t one.
( b ) : Let f€(s-Loc P o l Q )
and B ={(rl ( i ) ,
EQ
such t h a t
and g e Q("),
...,rn ( i ) 1 i es' ~ S A "
f ( B = g ( B , consequently
s'js
POT
r1,.-,r n
there exists a
f(rl
g6Pol Q
,...,rn )=g(rl,...,r
n )e9.
f e P o l &.(The proof o f ( a ) r u n s w i t h t h e same argument).
Thus
( b t ) r Let ~ ~ ( B - L OInv
C F) (m) and f e ~ ' " ) ,
rs I
.
89
and
...,rs
{rl,
B P I ,~...~,rs ,}SP t h e r e
s s g p ,consequently
s l s s.
For r1
,...,-
e x i s t s a C d n v F such t h a t
f (rl
,...,r9
) E C S ~ . Thus
YcInv F. ( ( a ' ) c a n be proved i n t h e same manner).
( c ) : Example: f : J -
W:xwx+l
€(Lot S m ) \ S m .
I
Another c h a r a c t e r i z a t i o n o f t h e o p e r a t o r LOC i e g i v e n i n
t h e n e x t p r o p o s i t i o n s . F i r s t o f a l l we need t h e f o l l o w i n g
1 .1 2 D e f i n i t i o n .
7
_--____--
Recall, a set
r e c t e d (upwards) i f f o r a l l
of s e t s i s called
X,Y er t h e r e e x i s t s a
----
di
Z
such that X V Y ~ Z .Now, let ue call a set
,...,
t
of sets to be
s - d i r e c ed (selN) if for all X1
XSET
--------------,
and
rl&X1,...,
..
~
that r , r
2. (In the
rseXs there exists a Z E such
terminology of ~.~ould&o68(p. 47211, 7 has the s-ary c o n
-
tainment
property.)
1 .1.3~ro~osition(cf.[~o68(~h.l .l 11). Let
-
Qs R A
--
be closed
-
under arbitrary intersections. Then the following holdsr
LOC Q ={~?1$fr5
Q and
(i)
7
is directed].
(ii) S - L O C Q = ~ U ~ " /0 f 7 " ~
and~7 is s-directed],stlN.
Thus Q is closed under (s-)LOC iff
-
&
-
is closed -under
-
ta-
-
king arbitrary unions of (s-)directed systems of relations
of Q.
-
2 particular, I-LOCQ
closed under arbitrary unions.
for finite B S T there
P
,-----r o o f . (i):w~ltrLet~ILOCQ.Then
exists at least one 6EQ with BfcrSy , and we can define
6jB:=njb~~
( BLCS~).
r={rB1BSP, B finite] is a directed system of elements of Q
(since 6BubBft
eBvBf).
Now, since
9 = u $ ~ I B~ finite]
~p,
9 , 9 is contained in the right side of (i).
lf+
l1
: Let
9 = V T ~ R for
~ ) a directed system
of Q and let ~ = { r,~
...,rt)~y
relations
(tclbi). Then there exist y1
.
ytE J such that rit gi(1 5 ift ) Because
exists a ~ € 7 Q5 such that
3 of
Bg
pl v
.,
,...,
is directed there
yt & s s p
. Thus, by
definition, QELOCQ.
5
: Let qes-LOC Q, B S p , \BIGS,
(ii):
{u$, 1
as above. Then
J=
B 6 y , ( ~ s$
( f is an a-directed system: in fact, for r e
...,rsa5
€3
...,rs123 we have {rl ,...,rs$ g ag q 7.
and B= {rl ,
B1
,
Thus 9 = ~ r b e l o n t~o s t h e r i g h t s i d e of ( i i ) .The o p p o s i t e
d i r e c t i o n c a n be proved a n a l o gdf o u s l y t o t h a t of ( i ) . I
1.14 P r o p o s i t i o n .
-
Let
be c l o s e d under a r b i t r a r y
-
Q=RA
-
i n t e r s e c t i o n s . Then t h e f o l l o w i n g c o n d i t i o n s a r e e q u i v a l e n t
(s€lN f i x e d ) :
(a)
Q = s-LOC Q .
(b)
BcQ
Lf
for all
-Proo-_f.
(and - c l e a r l y - o n l y i f ) p Q ( x ) 5B
X5B
(a)J(b)
w i t h 1x15 a.
f o l l o w s from
( H o t a t i o n c f . 1 .8)
B = u [ P ~ ( x ) /X g B , \ ~ \ g a ]by
1.1 3 ( i i ) and t h e f a c t t h a t a l l t h e
( b ) + ( a ) : If
system.
with
1x11 s
quently,
thus
7~ Q
r Q ( x ) form
i s s - d i r e c t e d and
t h e n , by d e f i n i t i o n , X g Z
r Q ( x )5
S-LOCQ f Q
f o r some
r Q ( z )= Z S V T = B . ~ y
by 1.13.
1.15 Remarks.
r e f l e c t those
an s - d i r e c t e d
X f B:=
UT
ZET,consew
( b ) we g e t U J E Q ,
I
The l o c a l o p e r a t o r s i n t r o d u c e d i n 1.9 w i l l
properties (of universal o r r e l a t i o n a l algebras)
which a r e caused by t h e f i n i t a r i t y of t h e o p e r a t i o n s and r e l a t i o n s under c o n s i d e r a t i o n . I f we c o n s i d e r a l s o i n f i n i t a r y
o p e r a t i o n s and r e l a t i o n s most o f a l l f u r t h e r r e s u l t s ($$4-6)
remain v a l i d by d e l e t i n g t h e s e l o c a l o p e r a t o r s ( c f . a l s o
[ K r / ~ o g , Doi8OJ )
The i n v e s t i g a t i o n of a l g e b r a i c s t r u c t u r e s by means o f
lllocalll p r o p e r t i e s (together with closure p r o p e r t i e s w . r . t .
composition) i s v e r y obvious and o f t e n used i n t h e l i t e r a t u r e ( c f . f o r i n s t a n c e such n o t i o n s a s l o c a l l y p r i m a l , l o c a l l y a f f i n e complete(cf.~~o/~al($5.5)l), i n t e r p o l a t i o n propert i e s f o r a l g e b r a s ( c f . [~ix], ~ ~ s ~ - / K / T ~ . ~ , [ H L I / w ~).] )
----oo-
.
$2
2.1
-
Clones of o p e r a t i o n s
Let u s r e c a l l t h e well-known n o t i o n of a c l o n e : A s e t
------c l o n e o
--f o- q------------e r a t i o n s
F c-O A i s c a l l e d a
notation:
(i)
F f OA
F contains a l l
(ii) For
...,xn-I
a l s o belongs t o
( i ~ n riel)
,
de-
) = x i ; and
..., f n ~ ~ ((n,m671~)
m)
t h e operation
F
.
F o r F g O A , t h e c l o n e g e n e r a t e d by
A
e;
( ---c o m ----------q o s i t i o n ) d e f i n e d by
g(fl,...,fn)
<F&
A,
,
p --r o j----------ecCions
n
ei(xo,
gcF ( n ) , f l ,
f i n e d by
on
, shortly
F
i s denoted by
(F).
T h i s i s t h e l e a s t c l o n e o f o p e r a t i o n s c o n t a i n i n g F.
The s e t
o f a l l p r o j e c t i o n s i s a c l o n e c o n t a i n e d i n every clone. The
~P(F> are also called
With
F<SA (or
we d e n o t e t h a t
F
B u Q --e rq------------o sit ion s
---
of
F
.
F$OA(1 1)
i s a subgroup (subsemigroup w i t h u n i t )
2.2 Remark.
------
There a r e some o t h e r e q u i v a l e n t d e f i n i t i o n s of
a c l o n e ( c f . e.g. f~chm.~.]); e.g., one c a n c o n s i d e r t h e
2
f u l l f u n c t i o n a l g e b r a P~=(O~;~~,~,S,A,~>(C~.
m a 6 6 , 7 6 7 , n ~ / ~ a lwhere
.
f, 'C o r A c a n produce any permut a t i o n o r i d e n t i f i c a t i o n o f v a r i a b l e s o f each f & O A and 0 i s
a s p e c i a l composition of two f u n c t i o n s ) and t h e c l o n e s a r e
By t h i s way one can avoid
e x a c t l y t h e s u b a l g e b r a s of
eA.
('2 3
the i n f i n i t e l y many s u p e r p o s t i o n o p e r a t i o n s i n 2 . l ( i i ) .
2.3
-
S A a n d O A ) form a group and semigroup, r e s p . , w i t h r e -
s p e c t t o composition. For
F5SA or
FSOA
t h e subgroup
1
o r subsemigroup (with u n i t eo) g e n e r a t e d by F w i l l be denoted
(FAA
,
o r <F>
respectively.
OA
For f i n i t e A ( b u t n o t i n g e n e r a l ) we have ( F ) ~ =<F>
)
A
O
A
f o r F c-S A *
Now we a r e a b l e t o g i v e a n Ifinnern c h a r a c t e r i z a t i o n o f
T F ( 6 ) ( c f . 1 .8) which does n o t use t h e i n v a r i a n t s of P:
2.4 Proposition(fPij/~al(l.l .19)],[~r(§Y
FgOA
and
,Lemma 31.7).
Let
6 e R A . Then ( c f . 1 . 8 )
r F ( ~ =) { g ( r l , - - , r n )
(
I
g t < ~ > ( " ) , { r ~, . * * , r n j ~ a , n ~ ~
For t h e sake of completeness we s k e t c h t h e well-known p r o o f :
--me--
Denote t h e r i g h t s i d e w i t h
g.
One e a s i l y checks t h a t $ i s in-
v a r i a n t f o r a l l f c F , and because {F)
we have
Inv F
69x, t h u s
implies
TF(o)Sr
g r l , , rn6
. At
contains a l l projections
t h e o t h e r hand,
,i
e
ggItF(*).
Pp(e)E
I
The f o l l o w i n g two lemmata show t h a t t h e c l o n e g e n e r a t i n g
p r o c e s s and t h e l o c a l c l o s u r e do n o t change i n v a r i a n t r e l a t i o n s and t h a t t h e l o c a l c l o s u r e p r e s e r v e s t h e p r o p e r t y o f
being a clone,
2.5Lemma.
(i)
(ii)
(iii)
Q5RA and FSOA. Then
&&
of o p e r a t i o n s , i . e . ( P O ~ ~ =QPolAQj
)~
A
s-Loc(F} i s-a c l o n e of o p e r a t i o n s (selt?);
LOC(F)
is a clone of o p e r a t i o n s .
PolAQ i s a c l o n e
P
roof.
-------
( i ) : The composition of f u n c t i o n s p r e s e r v i n g prQ
a l s o p r e s e r v e s p. ( i i ) : Let ges-Loc F'"), f l , ,fries-Loc F (m),
m
~ = { b ~ b t f t A , t s s , and Bf:={(flbi,...,fnbi)llCif
tJ.
...
,...,
Then t h e r e e x i s t
fi ,...,fl;cF(")
gt&~(m),
~ J B ' ,f i ( B = fiIB (1 J 16 t ). Therefore
with
g (fi
...,fA)
,
on
such t h a t g t l B 1 =
,..., f n ) c o i n a i d e s
g ( f l ,...,fn)ts-Loc<F>.
g(fl
B , consequently
( i i i ) : C l e a r l y , w i t h s-Lot@ a l s o t h e i n t e r s e c t i o n Loc(F>
a l l these i s a clone.
2.6 Lemma.
(i)
F
m
OA.
Then we have:
= 1nvP1(F)=
1nv$)F
for
-
(ii)
Let
of
1nv?)Loc{F)=
~nv~)s-Loc(F)
-
I&rnfs~N;
InvAF = lnvA(F) = InvALoc(F).
P
------roof.
( i ) : Let 9 6 ~ n v p ) ~Because
.
t h e s e t s i n ( i ) must
form (from l e f t t o r i g h t ) a d e c r e a s i n g c h a i n (cf.1.7)
f i c e s t o show g ~ l n v ( m ) s - ~ oI?c
i t suf-
. C l e a r l y P e ~ n v ( m ) <(~s>i n c e su-
p e r p o s i t i o n s of F a l s o p r e s e r v e q ( 2 . 5 ( i ) ) ) . Now, l e t f e
s - L o c ( ~ ) ( ~ )and rl
,., ,rncp.
By d e f i n i t i o n o f t h e .-local
sure(1 .g , n o t e mgs), t h e r e e x i s t s a
,
)
( i i ) i s a d i r e c t consequence of ( i ) .
1
...,
such t h a t f (r, ,
...,rn , consequently f ( r l ,..., r n ) ' ~ , i . e . ,
9 . Thue g r I n v s-LOC(F).
rn) = g(r,
serves
$e{F>
clo-
f pre-
(3.1 )
$3
Clones of r e l a t i o n s
3.1 M
otivations.
-----------
There does n o t e x i s t a f i x e d n o t i o n o f a
ltclone o f r e l a t i o n s t t . Something l i k e a n c l o n e t t i s g i v e n w i t h
~ u z l i n ' s t h e o r y of pro j e c t i v e s e t s and, moreover, t h e r e a r e
c l o s e c o n n e c t i o n s t o t h e t h e o r y o f c y l i n d r i c a l g e b r a s (Tars k i ) , p o l y a d i c a l g e b r a s ( H a l m o ~ )and t h e a x i o m a t i z a t i o n o f
enlargements ( R o b i n s o n ) ( I wish t o thank P r o f , J,Schmidt f o r
t h i s hint)..
The d e f i n i t i o n we a r e going t o p r o v i d e now s h a l l s e r v e as an
analogon t o c l o n e s o f o p e r a t i o n s w i t h r e s p e c t t o t h e G a l o i s
c o n n e c t i o n Pol-Inv, Theye a r e two approaches t o g e t canonio a l l y t h e n o t i o n of a c l o n e o f r e l a t i o n s :
a)
The c l o n e o f o p e r a t i o n s g e n e r a t e d by F c OA c o n s i s t s
o f a l l term f u n c t i o n s ( i . e , superposition^ o f F) of t h e a l gebra < A ; F > ( c.[~r],
~
2.1 ) T h e r e f o r e , f o r a r e l a t i o n a l a l g e b r a <A;Q)(Q 5 R A ) one c a n t r y t o c o n s i d e r !!term r e l a t i o n s "
which a r e b u i l t up from t h e e l e m e n t s o f Q by u s i n g t h e
( s e t t h e o r e t i c ) composition o f r e l a t i o n s ( i n c l u d i n g such
operations l i k e permutations, i d e n t i f i c a t i o n s o r d e l a t i n g of
v a r i a b l e s ) , If we c o n s i d e r r e l a t i o n s as p r e d i c a t e s ( o v e r A )
t h e n t h i s means t h a t f i r s t l y we have t o t a k e a s e t
of
formulas ( w i t h some p r e d i c a t e symbols) and secondly we must
define:. The c l o n e g e n e r a t e d by Q i s t h e s e t of a l l p r e d i c a t e s which a r e d e r i v a b l e from elements o f Q by means o f
formulas i n 8 ( " i n t e r n a l d e f i n i t i o n t t o f a c l o n e ) , E , g . , t h e
composition of r e l a t i o n s l e a d s t o t h e s e t
of a l l existent i a l formulas w i t h o u t n e g a t i o n or d i s j u n c t i o n . C l e a r l y , t h e
c h o i c e o f 5 should depend on o u r a l g e b r a i c purpose, i . e . on
t h e q u e s t i o n : what we want t o do w i t h c l o n e s of r e l a t i o n s .
Roughly s p e a k i n g , t h e above mentioned f o r m u l a s a r e s u i t a b l e
f o r d e s c r i b i n g g e n e r a l systems o f o p e r a t i o n s , I n s p e c i a l cas e s however, namely f o r unary ( o r b i j e c t i v e ) o p e r a t i o n s t h e
s e t B must be extended by a l l o w i n g a l s o d i s j u n c t i o n ( o r d i s j u n c t i o n s and n e g a t i o n s ) o f formulas.
.
-
We w i l l n o t develop a formula c a l c u l u s h e r e because
equiv a l e n t l y (cf.~o/Kal(§2.1)])
we u s e some o p e r a t i o n s on RA
i n s t e a d o f formulas.
-
b)
For a moment l e t u s f o r g e t t h e f i n i t a r i t y o f t h e r e l a t i o n s under c o n s i d e r a t i o n . Then one can prove (cf.[~os72,75],
n ( r / ~ o g ) t h a t t h e c l o n e s of o p e r a t i o n s (F) are e x a c t l y t h e
G a l o i s c l o s e d s e t s P o l I n v F ( F g O A ) . I n o t h e r words we have t h e e x t e r n a l d e f i n i t i o n : (F) i s t h e g r e a t e s t s e t F ' of
o p e r a t i o n s f o r which (A;F)"
and <A;F')"
have t h e same suba l g e b r a s f o r a l l n ( n c a n be i n f i n i t e , t o o ) , c f . 1 . 5 ( i i ) .
The c o u n t e r p a r t ( i . e . t h e d u a l w i t h r e s p e c t t o Pol-Inv,
c f . 1 - 2 ) of t h e n o t i o n t t s u b a l g e b r a Mi s t h e n o t i o n ttpolymorphism". Therefope t h e polymorphisms should p l a y t h e c e n t r a l
r o l e f o r r e l a t i o n a l algebras. This motivates t h e following
" e x t e r n a l d e f i n i t i o n t t ( i n f u l l analogy t o c l o n e s of operations) :
The c l o n e o f r e l a t i o n s [Q] g e n e r a t e d by Q g R A i s t h e
g r e a t e s t s e t Q f o f r e l a t i o n s f o r which A =(A; Q) and A ' =
{A; Q') have t h e same n-nry polymorphisms (i.e. ~ o r n ( ~ =~ , ~ )
H Q I ~ ( A ' ~ f, o~ r) a) l l n. I n o t h e r worde, [Q] should be e q u a l
t o t h e G a l o i s c l o s e d s e t Inv P o l &.
-
-
C l e a r l y , t h e n o t i o n o f a c l o n e depends on t h e range o f n ,
i . e . whether we a l l o w t o c o n s i d e r i n f i n i t a r y o p e r a t i o n s ( o r
r e l a t i o n s ) o r not. Because we want t o d e a l w i t h f i n i t a r y r e l a t i o n s and o p e r a t i o n s o n l y , we s h a l l s e p e r a t e t h o s e propert i e s o f t h e G a l o i s c l o s e d s e t s of r e l a , t i o n a which a r e caused
by t h e f i n i t a r i t y o f t h e o p e r a t i o n s . For t h i s r e a s o n we i n t r o d u c e d t h e l o c a l o p e r a t o r LOC ( a n a l o g e o u s l y Loc f o r oper a t i o n s ) which d e s c r i b e s t h e i n f l u e n c e of & f i n i t a r i t y w h i l e
t h e n o t i o n o f t h e c l o n e t o be d e f i n e d i~t h e e s s e n t i a l p a r t
of t h e G a l o i s c l o s u r e (and works a l s o i n c a s e o f i n f i n i t a r y
relations).
Thus we t r y t o a v o i d any i n f i n i t a r i t y i n t h e i n t e r n a l def i n i t i o n of c l o n e s o f r e l a t i o n s and we s h a l l d e f i n e t h e c l o n e s by s u i t a b l e chosen s e t t h e o r e t i c o p e r a t i o n s on f i n i t a r y r e l a t i o n s . (However, t h e a t t e n t i v e r e a d e r w i l l f i n d
t h a t , i n f a c t , we cannot escape from a c l a n d e s t i n e use of i n finitary relations
because we need i n f i n i t e many J-cpantifiers
and probably one can prove t h a t t h i s has t o be.'
-
-
Now we a r e going t o d e f i n e ( s e t t h e o r e t i c ) o p e r a t i o n s on
RA some of which w i l l be used t o d e f i n e t h e c l o s u r e proper-
t i e s of clones.
3.2 D e f i n i t i o n s .
-
(RO).
Diagonal r e l a t i o n s :
The r e l a t i o n s
P
6,e R?),
m e l N and T i s an equivalence r e l a t i o n on
by
where
g, defined
,-
EmL = { ( x o , b b . , s - l ) e ~ mI ( i , j ) t . - r + xi = x j )
a r e c a l l e d t---------r i v i a l o r d-i-g_@;o
-----n a l relations.
Let
DA be t h e s e t of a l l diagonal r e l a t i o n s t o g e t h e r with
t h e empty r e l a t i o n @ The elements of R A \ DA a r e c a l l e d t o
be
.
n
ontrivial.
---------------
(R1)
S u b s t i t u t i o n s :
_-________________Poramapping
PtRn(m)
-
X : n--+g,
we d e f i n e X ( p ) c ~ p and
)
and VCR?)
x-' ( a ) t R A(m 1
as f o l l o w s (n,melN):
contravariant substitution functor:
I
1
7rtq) := ( a n ( o ) , *.- , a n(n-1 ) )&A" ( a o , *.. , a m-1 ) c Y ]
,
covariant substitution functor:
z-'
( B ) :={(ao,
By a s p e c i a l choice of
(Rla)
P
ermutation
---------------Take
f ( 9 ) for
..., am-1 ) E A ~ I
(aX(0),
...,d ~ ( n - 1) ) ~ d .
we o b t a i n many well-known o p e r a t i o n s
T
o
f c---------------oordinates :
--X:
-n -neSn;
(R1,b) D
---------e l a t i n ~
o
f ---------------coordinates :
--For ( m - I m
%(ri) = # a 0
9.-
-
i
we g e t
1 3a : (ao
, a m-2
,am-2,a)tgl;
,.**
o r more g e n e r a l
(Rlc)
---- .-------------..-.-------
Pro_S_ections o
nto coordinate's :
-----For i n j e c t i v e T : fi -m:
(Rld)
-
-
i wji ( i e n ) we g e t
D
---------o u b l i n ~
o
f coordinates :
---
------------.---
For
-
~ t ( m + l -+g:
)
i oi (iern), rn-rn-1,
we g e t
n ( p ) ={(ao,..* ,a*-I
(Rle)
--.
I.....................
d e n t i f i c a t i o n of coordinates :
--c--------------
For R : 2
4 (n-1
-
): i)
I
,i ( i € n - 1 ), n-1 b n-2,
f i c t i v e c---------------oordinates
--- -------- -.---------
(.Rlf), A d J o i n i n g
For
F:
-n ---+(n+l)
: i
w
i
, we
we g e t
:
get
Note t h a t t h e d e f i n i t i o n of s u b s t i t u t i o n s keeps i t s s e n s e a l s o
f o r i n f i n i t e ( o r d i n a l numbers)! n
___---_-__----_--_
For
cmR2), I n C e r a e c t i o n :
i s t h e i n t e r s e c t i o n of a l l
(R3)
and
m
.
yiit~p),
icI,
Ti,
ie1.
____ .-__-______
For qt~p),6c~p),
t h e composi-
Composition:
tion
~
o
i sr t h e following (m+n-2)-ary r e l a t i o n :
90
= {(:aO,
...,am+n-3 I
)
&&A
: (a0, .r ,am-2,a
e p and
(a9 am-l 9 **9am+n-3) e c
(For m=n=l we d e f i n e
FOG=
$4 ).
---------- --s u ~--e r ----------g o s i t i o n:
(R4)
General
Ti : mi
-4
o
9
(mi
For picRA
,
(N a r b i t r a r y o r d i n a l number, e ={P I ~ < w) ],
o
i r I ( i n d e x s e t ) , and :
- 4III
we d e f i n e
t o be t h e r e l a t i o n
R(m-1 )
For
pica, t h e r e l a t i o n s
:For a l l i c I
4,,-) (9i)
-
are called
( g e n e r a l ) s u p e r p o s i t i o n s o f (elements o f ) Q .
Remarks: We can t a k e t h e i d e n t i t y (i c-, i ) f o r 7r i f we a l l o w
s u b s t i t u t i o n s of coordinates.
With t h e n o t a t i o n s g i v e n i n (R1) and (R2) we have
-am----
Note t h a t t h e
CsR5)
/
_______s--_
u ~-__
e r ---_------p o s i t i o n :
gpecial
Gzi
/
/
p i need not be d i f f e r e n t .
let
1,
mappings 7 : & 3
For i n f i n i t e A ,
be t h e s e t o f a l l monotone i n j e c t i v e
5 , nL IN. For a family Q = ( P ~ ) ~ ~ ~
n
o f r e l a t i o n s ( w i t h % E R ~ )f o r r c I n ) we d e f i n e t h e
rn-special s u p e r p o s i t i o n of
Q
as f o l l o w s :
3.3
-------
3.4
Notations.
Remarks.
It i s easy t o s e e t h a t t h e o p e r a t i o n s ( R 1 ) ( ~ 3 ) , ( ~ 5a r)e s p e c i a l c a s e s of (R4)
one h a s t o s p e c i a l i z e
c,T,Ti, I. The g e n e r a l s u p e r p o s i t i o n was d e f i n e d and s t u d i e d
a l s o i n B z 7 8 3 by means of so-called formula schemes which
i n e f f e c t a r e nothing e l s e t h a n t h e t r i p e l ((yi,7t'i)ieI,d,V.
-
-
-
d e r e d a s functio.ns
a:n -+A:ioa(i).
t h e composition of
- is
a = ( a ( i ) ) i p n € ~ n can be consi-
The elements
and
a n element o f
For
- notation
a
Z:
g-+n
-
w a ( f i r s t R. t h e n a )
Ame Thus t h e n o t i o n e
i n 3.2 can be
d e f i n e d s h o r t l y as f o l l o w s :
1
( % ) i e 1 = { ~ aa e k w i t h
Definition.
-------------
r e l a t i o n s
A set
on
Q
is called a
~GI).
------- o-__
f
clone
A
- notation:
if
QgRA
Riac pi f o r a l l
Q$RA
-
contains the t r i v i a l r e l a t i o n s
0
and
AeRA
and i s
c l o s e d w i t h r e s p e c t t o g e n e r a l s u p e r p o s i t i o n (R4).
For a r b i t r a r y
Q (i.e,
by
Q.ERA, t h e clone ( o f r e l a t i o n s ) g e n e r a t e d
t h e l e a s t c l o n e c o n t a i n i n g Q) w i l l be denoted by
There a r e some e q u i v a l e n t d e f i n i t i o n s ,
3.6 P r o p o s i t i o n .
Let
e q u i v a l e n t ( c f . 3.2,
(i)
Q
-
QgRA. The f o l l o w i n g c o n d i t i o n s a r e
3.3):
-i s-a c l o n e of r e l a t i o n s :
Q
5Rg
.
-
-
all
- Q contains trivial relations),(Rl),(R2),(R3),(~4) and (R5).
closed with respect to (RO)(i.e.
(ii) Q
is closed with respect 2 (Ro),(R1),(~2),(~5).
-
(iii) Q
-
(x t 2 k )
(iv) Q is closed with respect to (RO),(Rla),(Rle),(R2),(R5).
For finite A,
-
------
(i) is also equivalent to each of the folloI _ - -
wing conditions:
(v)
Q
(vi) Q
(vii) Q
is closed with respect to (R1 ) ,(R3) and contains A.
is closed with respect to (RO),(R~~),(R?~),(R~).
is closed with respect to (RI a), (Rle) ,(R3) and
contains
(va)
8
(where E ={(0,0),(0,1)
,(I ,O), (1 ,I 1, (2,2$.
is closed with respect to (Ela), (Rl e l , (Rlf) ,(R3)
& and contains
-
A E R A(1
.
P
------.
r o o f . We give some hints only and will not go into technical details. The construction ( ~ 4 )
can be transformed into
the form (R5) using (RO), (R1 ) and ( R 2 )
Every diagonal can be obtained from
while
- and vice versa.
Arne D?)
using (R1 ) ,
ACRA ) generates A~ via (Rlf). All substitutions (R1)
can be generated with (Rla) and (Rle) using (RO) and (R3)
(the latter is derivable from (R5)).
For finite A, the ope-
rations (R4) can be reduced to finite
(Lemma Z)])
I and ac (see f~z78
and can be expressed by (RO),(Hl) and ( R 3 )
[~b'/~al('l.I .9)]).
(see
(a)
3.7 Remarks.
(i) The set
.------
DA of all diagonals is a clone
of relations contained in every clone.
(ii)
Clones of relations can be considered as the subalgeof the algebra <~~;&,r,f
,A,v, n,
m& Dl of type
(p) >
5 (7).t(p) = R2(9)
...
for
pc~p)
and t h e p e r m u t a t i o n s
TI =
n
(01
n - I ) , 7C2= ( 0 l ) o f SA, A = ( R l e ) , V = ( R I ~ ) , d e n o t e s
t h e i n t e r s e c t i o n o f a f a m i l y bf r e l a t i o n s ) o f c a r d i n a l i t y 21A'
and
i s d e f i n e d a s i n 3.2(R5). The proof f o l l o w s from
3 . 6 ( i v ) and t h e f a c t t h a t
g e n e r a t e a l l (Ro) and
(RI'). For f i n i t e A , t h e above a l g e b r a can be chosen much
s i m p l e r : The c l o n e s a r e e x a c t l y t h e s u b a l g e b r a s of t h e a l g e b r a (RA; A , ~ , T , A , ~ , Oof) t y p e <0,1 ,I ,I ,1 ,2) ( c f , B 3 / ~ a l ( ~ . 4 3 Y ) .
nm
&,r,'~-,A,v
(iii)
From t h e p o i n t of view o f l o g i c , c l o n e s o f r e l a t i o n s
a r e t h o s e s e t s o f p r e d i c a t e s (on A ) which a r e c l o s e d under
f i r s t o r d e r f o r m u l a s c o n t a i n i n g > , A , = but n o t V , ~ , Vand
,
u n d e r i n f i n i t e i n t e r s e c t i o n s and 3 - q u a n t i f i c a t i o n s (i.e.
we can u s e p o s i t i v e f i r s t o r d e r f o r m u l a s w i t h i n f i n i t e many
e x i s t e n t i a l q u a n t i f i e r s and conjunct i o n s ) , c f . [ K r / ~ o i J .
I n . analogy t o 2.5 and 2.6 we have t h e f o l l o w i n g p r o p e r t i e s .
3.8
&&
Proposition.
Q 5 RA
and
F 5 QA.
---
Then
(i)
InvAF i s a c l o n e of r e l a t i o n s , i . e .
(ii)
s-LOC[Q~
i s a clone of r e l a t i o n s ( s c B).
( i i i ) LOC [Q]
-------
p r o o f.
--i s-a c l o n e of r e l a t i o n s .
(cf
[ J ~ V ~ F=] InvAF
.
1 .9)
7i
( i ) : A(Ti) ( F ~ ) ~ 3.2(R4))
~ ~ ( c p~r e s e n e s a n opera-
t i o n f e O A whenever each
9
preserves f
( i i ) and ( i i i ) w i l l f o l l o w e.g.
.
from theorem 4.2 but t h e
proof c a n be done e a s i l y a l s o by checking t h e d e f i n i t i 0 n s . l
3.9 Proposition.
(i)
Let
-
QfRA.
Then we have:
---
P ~ I ~ =) PQO I ~ CQI=
)
P ~ ~ ~ ) M c P~O Q
~(~
I )=S - L O C
for
I
.
I i-n f a G B .
( i ) r Let
.----I-
P r o o f.
f c P o l ( " ) ~ . Because t h e s e t s i n ( i ) must
form a d e c r e a s i n g (from l e f t t o r i g h t ) c h a i n i t s u f f i c e s t o
show t h a t
f e P o l (")S-LOC [Q]
p e r p o s i t i o n ~o f
Q
. C l e a r l y f c ~ o l ( ~ ' [ becauae
~]
su-
are also invariant f o r
f ( e a s y proof o r
,...,r n s y . By 1 .9
( n o t e n ~ s ) ,t h e r e e x i s t s a c r c l ~ ]such t h a t jrl ,...,rn],cssp.
Consequently f ( r l , . . . , r n )&@SF, i . e . f p r e s e r v e s p .
uae 3.8(1),).
Let
~ L B - L O C [ Q ] (m)
(ii)immediately f o l l o w s f r o n ( i ) .
54
The G a l o i s c o n n e c t i o n
Pol
and
rl
I
- Inv
The f o l l o w i n g two theorems a r e b a s i c f o r a "General G a l o i s
Theory o f o p e r a t i o n s and r e l a t i ~ n s ~ ~ ( icnft .r o d u c t i o n ) because
t h e y c h a r a c t e r i z e t h e G a l o i s c l o s e d s e t s of o p e r a t i o n 8 and
r e l a t i o n s , r e s p . The p r o p o s i t i o n s 1.11, 2.5 and 3.8 s u g g e s t
t h a t t h e s e G a l o i s c l o s e d s e t s might be e x a c t l y t h e l o c a l
closed clones, i.e.
, the
l l e x t e r n a l l l and l l i n t e r n a l l l d e f i n i -
t i o n s of c l o n e s might c o i n c i d e ( c f . 3 . l b ) .
4.1 ~ h e o r e m ( c f [. ~ . e u, D a / ~ i x ( L e m m a 3.1. )]
Let
-
,[~orn77:bJ).
F r OA. Then w e have:
= PolAInvAl"
.
(a)
LOC(F>
(b)
s-LOC(F>
= ~ o l ~ ~ n fvo ri s~6 N
) ~( f o r S=I c f . [ ~ c h m ~ .
(Thm.1 .6y).
4.2 Theorem.
Let
-
---
Q g R A . Then we have:
(a)
L O G [ ~ ] = InvAPolAQ ( c f . LSz78 (Lemma 4 ) I , f ~ e i(p.99 ) I ) .
(b)
s-LOG
[QJ = ~ n v ~ ~ foo r l s~6 B~ (for/*efl
) ~
-
c f .[Go68]).
H M q q
-
ik~~g.au&
-------
Remarks.
R e s u l t s c o n c e r n i n g t h e c h a r a c t e r i z a t i o n of Galoia
c l o s e d s e t s ( w , r , t . Pol-Inv). o f r e l a t i o n s c a n be found
w i t h more o r l e s s m o d i f i c a t i o n s and r e s t r i c t i o n s a l s o i n :
-,[sz78]
( I n v Pol Q f o r f i n i t a r y r e l a t i o n s and
operations ) ;
[ K F / P O ~ ( I n v P o l (2 (conlgebre's de P o s t ) f o r i n f i n i t a r y r e l a t i o n s and o p e r a t i o n s ) ;
[ ~ o / ~ a l ] , [ P o / ~ a d ( I n v A P o l g ~( P o s t c o a l g e b r a s , R e l a t i o n e n a l g e b r e n ) , ~ n v ~ ~ )Q,
o l InvAAutAQ
y
(Krasner a l g e b r a s ) f o r
f i n i t e A);
[kr6€3],[Kr76i]
( I n v Aut
I n v p o l ( ' )Q);
[pa731 ( I n v P o l Q f o r o p e r a t i o n s and r e l a t i o n s d e f i n e d
on a f a m i l y o f f i n i t e s e t s ) ;
[~os78] ( 1 n v ( " ) ~ o l Q , c f . 10.5);
[~a/st77b] (End P o l f , f c O A , n o t e ( ~ n Pdo l ) * = ( O i l )'o
I n v P o l );
[ ~ a / ~ t 7 7 c ] (End P o l S , S SO'!
I);( c f . a l s o De])
[ ~ a / ~ t 7 8 ] (Pol End F , F ~ O ~ ) .
-
-
(
a,
?
P r o o f - of 4.1:
dm-------------
(b): B y l S 7 a n d 2 . 6 w e h a v e
s-Loc{F> 5 P o l I n v s-Lot<@ g P o l ~ n v ( ~ ) ' b - L o c ( F =
> Pol ~ n v " ) ~ .
To show t h e o p p o s i t e i n c l u s i o n l e t
f~s-LOC<F).
Let
B ={bo,
ft~ol(~)~nv(~
We) F
prove
.
...,bt.,}~An, t g
s, ri:=(bo(i),
...,
b t - l ( i ) ) , i ~ g and
,
6 = ( r i l i ~ g fS. i n c e T ~ ( @ ) ~ I ~ v ((and
~)F
f c P o l lnv(')F f : ~ o Il ~ v ( ~ ) Pt h) e r e e x i s t s ( c f . 2.4) an ge<F>
such t h a t
f ( r o , ...,r n-1 ) = g ( r 0 , , n-1 ), i.e.
f~s-LOC(F).
( a ) f o l l o w s froms ( b ) s i n c e
- r\ P o l
stm
LOC
f)~=g/~,hence
.
@>=
SGJN
-
U
~ n v ( ~ ) F = ~ ol nl v " ) F
s € IIq
= P o l I n v F.
s-LOC<F>
Proof
o f 4.2:
s-LOC
[Q]
s Inv
( b ) : By 1 . 7 a n d 3.9 we have
-----_______L__
NOW,
POI.
s-LOC
[QI5 I n v POI. (S)S-LOC[ C J ~=
I ~ VP
OI.(~)~.
f o r g t ~ ~ ~ ( m ) ~( m~a m
l )( , swe) ~
a r e going t o prove
pe
p = U T f o r t h e s - d i r e c t e d (1..12) sysI B g p , lB($s] where F = pol(')^. B y t h e n e x t
a-LOC[Q]. Note t h a t
T=IT,(*)
tem
lemma (4.3b) we have
9 = U T Es - L O C ~ ] ( c f .
T ~ ( B ) L ; [ Q ] ,t h u s
1 . 1 3 ) and we are done.
(a) f o l l o w s from ( b ) s i n c e
LOC 101=
0 s-LOC[Q]=
El c
2
T
The hard c o r e o f t h e proof of 4.2 i s t h e f o l l o w i n g lemma.
4.3 Lemma
-
For
-
( o f .[~z78(Lemma 21-7).
-
Q c R A and
F = PolAQ
w e have:
L
_
_
--
a)
I ? ~ ( D ) E [ Q ]f o r a l l f i n i t e
b,
rF(s)
-------
( B ) E ~ f] o r a l l
B &A"
B $A"
(ncx).
with J B J $ s ( s , n c B ) .
(B) f o r
b ) f o l l o w s from a ) s i n c e P F ( D ) = 1
F(s)
P r o o f.
l ~ l g o(cT. 2.4).
a ) : Extending t h e proof g i v e n i n [ ~ o / ~ a l ] f o r f i n i t e A t o
- what
infinite A
was a l r e a d y done i n d e p e n d e n t l y by L.Szab6
(~2781- and f o l l o w i n g [ ~ z 7 8 ( ~ r o o fof lemma 2)] we c o n s t r u c t
a
F~E[Q]
Let B=fb0,
and show
.0
.
,b,-,]g~
vB=pF(li)(for f i n i t e B ~ A " ) .
n
,
Ip
For p ~ ~ ( m ) ( m sl le )t
a = (rj(i))(i,j)cmME9
-
-
. = z1
. ( E ) ! : = ( b o ( i ) , me ,bS-, ( i ) ) e ~ ' ( i e n ) .
be t h e s e t of a l l m a t r i c e s
t h e rows o f which we denote by
..., r g - l j S ~( i . e .
( r 0 ( i ) , r s-1 ( i ) ) ( i c z ) , such t h a t fro,
colwns belong t o 9 ) For a l l Y C Q ("') and
.
z~(M)=
M
B
the
Iy we d e f i n e
the mapping
:?!-As:
jr;
h :
let
n
---;r
As
:
it-+zi(hl)
(icm),
- and
i w zi(.U)s
(irn).
-
Then, by 3.5,
(azO(M)
,*.. 'm-1
I nCAAs with
i.e.(3.4)WB=t%s
(M )cp for all Me$
)I
mcm
'3,
G a b p for all hlcIp, *EQ),
belongs to [
.
]
Q
We are going to show tkB=Tp(Bj.
We observe: If
a=(a
fulfilles the condition
z&AS
M a t ? for all McF and pc Q ,
(*1'
2
)
?
z W az preserves all phQ. In fact,
then f: As - A :
...
... ,rs-1) g P
M
implies f (rO, ,rs-1 ) =RgaEP.
Conversely,
f : A~ -+A preserves all p E Q , then a = (f(z))ZEAS ful-
M= fro,
if
, p aQ (m
fillea (*).Therefore
rB={(a
0
'
={(f(zO)
,
--
I
1 (az)zeA8 fulfilles ~ ~ *=) ~ f
'n-I
,*** ,f(zn-l
f E F = Po1 93. Thus, by definition
,a
))I
of the zi and 2.4, we get ~J={f(bo,-.,b,-l)lfc~I=TF(B).
ff
The following propositions which are corollaries to 4.1
and 4.2 give the characterization of clones of operations
and relations, resp., via the Galois connection Pol-Inv.
4.4 Proposition. For
-
F g OA, the -follow-
--
conditions are
equivalent :
(i)
-
F L O A (i.e. P = (F)~
(ii) P = PolA InvAJ?
,
-
) and
A
Loc F = F
,
proof.
-------
(i)+(ii)
(iii)+(i)
by 2 . 5 ( i ) and I .I I ( a ) . I
4.5 P r o p o s i t i o n .
-
by 4 . l a ,
For
-
(ii)+(iii)
QCRA,
obvious,
the f o l l o w i n g
-
conditions are
equivalent :
QsRA
(i.e,Q=CQI)
(i)~
( i i ) ! Q = InvAPolAQ ,
(iii).
IF2 0 ~ Q:
= InvAF
p r o o f.
------_
i
(iii)+(i)
by I .I1 ( a 1 ) and 3 . 8 ( i ) .
) by a
R~
and
-
LOC Q = Q
,
.
, (ii)+iii)
obvious,
B
Moreover, 4.4 and 4.5 answer t h e q u e s t i o n u n d e r which
conditions a s e t
Inv F f
F
or
respectively.
ia representable a s
Q
Pol
Ql
or
Such c o n c r e t e c h a r a c t e r i z a t i o n problems
w i l l be t r e a t e d i n t h e ' n e x t paragraphs. I n p a r t i c u l a r we
have f o r t h e group c a s e :
4.6 P r o p o s i t i o n .
-
For
C-
-
G g S A , ,I&f o l l o w i n g c o n d i t i o n s a r e
equivalent :
(i)
LOC(G)~
Gr.SA ( i . e .
(ii)
G = AutgInvAG
(iii)
39 5 R
~ :G =
= S A n Loc G = G.
A
F = (G)
,
)
S~
and
-
G = SAnLoc G
.
A U ~ ~ Q
(ii)+(iii)
obvious.
,
i s a subgroup of SA ( c f . 1.6a). More-
( i i i ) + ( i ) : G=Aut Q
over, f o r
f c S A f iLoc G
we have
f-'4 SA nLoc G ( t h i s e a s i l y
f o l l o w s from t h e d e f i n i t i o n s ) and by l . l l a
p r e s e r v e s a l l peQ. Thus
------
Remark: F o r
I n 4.6,
SA nLoc G = Au* P
G -I s A we have
-
4.7Lemma.
-
G =(G,)
(I )
OA
W
&
and
G = Loc 1
I
-
and G S S A w e h a v e :
-F o r F S-O A (LOC$>"
)
A
(LOC<G> )
c
s~ -
LOC
-.----.
(Loc(F))*=(Pol
.
( c f . 4.1,
(notation cf. 1 . 3 ) ,
LOC [F*]
Proof.
= LOC[F']
.
S A n L o c G =Loc
( i ) c o u l d be r e p l a c e d
(i)
every feLoc G
[GO]
.
Inv F ) ' f-( P o l
I i i i , 4.2).
f € S A we have < G > ~ ' f [G*]
A
o f t h e lemma.
P o l ~ ' ) ' c I n v P o l F"
Since
(f-"
)to
E [f*] f o r
what i m p l i e a t h e eecond i n c l u s i o n
Part 2
CONCRETE CHARACTERIZATIONS
OF RELATED ALGEBRAIC STRUCTURES
55
Concrete characterizations I.(Characterization of
operational systems via relational ones ) ,
In this paragraph we investigate the problem whether for
a given permutation group
G, a transformation semigroup H
or similiar "operational atructuresu does exist a relational
algebra $-=<A;Q>auch that
G = (w-)~ut8,H = ~ n d % or
"something like thisu respectively. All theee problems will
be covered by the following general problem:
5.1 -------------------------Concrete characterization problem:
-
I-----
Given a set
A and FicE.cOA
(iaI), does there exist a
1
relational algebra & =<A; Q)(Q
S RA) such that
Fi = Ei A POIAQ ?
Under which conditions one can chooae the relatione of Q
to be of bounded rank (i.e. of bounded arity) ?
Specializing Ei we get e.g. the following characterization problems :
(5*2)
yields the characterization of the
Ei =
automorphiems
weak
S~
oil
endomorphisms
0p)
n-ary polymorphisms
polymorphiams
OA
% =(A;
of relational algebras
Q)
.
Moreover, we can get a sinlultaneous characterization of
these structurese For automorphim groups we need l i k t l e modifications.
The answer to 5.1 is given in the following theorem:
5.2 Theorem.
Let F
-
--
-
i EiS
~ OA(irI index set) and F = V Fi.
There exists a relational algebra <A;Q)
-
Oo
QfRA Or
p)
QcR*(I ),u
... )';R
(SGTJ),
0
with
ie1
-
-.--
resp.,
auch t h a t
--
Fi = Ei n PolAQ (ioI)
if and only
I
_
if
-
)
Fi = E~OLOC{F> (icI) or
6)
Fi
=
Ei n a-LOC(F>
(id), resp.
If.
wewant to consider automorphisms Fi=Ein Aut Q
Remark.
---A
then LOC(F>
above must be replaced by Loc (1 - 1 { -
1
fftsA {f ,f0'f C ~ o {$
c (analogeouely for S-LOC(F)
P
-_.-_.r o o f . a)It5y1:Since FiSPol
Q
we
1.
have FCPol Q
and
(by 2 . 5 ( i ) ,11.1 1 a ) L O ~ ( F ) ~ L O C@=
(PC
Pol
I ~ Q o Thus FiS
- E~
6
pol Q = F~ ( n o t e F~ c
- ~ o c ( F ) s i n c e F~ s F )
fi
Take
11+11:
Q:=
Inv F. Then
.
E~~Loc(F)
Fi=Ein L ~ c ( F > = E n
~ P o l Q by 4.la.
f ) can
~ be proved analogeously ( u s i n g 4.1 b )
.
The remark f o l l o w s from t h e o b s e r v a t i o n ( c f . 4.6(remark))
that
Loc ('1-'){F>=
5.3 Remark.
------
~ u 1nv
t
F
.
w
Note t h a t f o r 5.2 case
P)
we could t a k e
to
Q
be a s e t o f s-ary r e l a t i o n s only because every r e l a t i o n
Q G
( ' ) ( i s 8 ) can be extended t o a g e R i s ) ( b y a d j o i n i n g f i c t i v e
R~
c o o r d i n a t e s , c f 3.2 (Rl f ) ) such t h a t P o l 9 1 9019
.
.
5.4 D e f i n i t i o n and n o t a t i o n s .
The a l g e b r a ( 0 p ) ; (et)ien, O)
of t y p e < ( ~ ) ~ , ~ , n +where
l>
o$g,fl ,..., f n ) :=g(fl ,..., f n ) ( c f .
of
2 , l ( i i ) ) i s c a l l e d t h e f__--_u l l -____
M e n --~ e r--a l g -__--e b r a
n-ary o p e r a t i o n a . Thue
F
iff
OA
--
Thia can be proven t o be e q u i v a l e n t t o
Therefore, f o r
Fs<F>An!
A
OA, F(") i s a subalgebra of t h e Menger
--
i n p a r t i c u l a r , F(' ) i s a subsemigroup of
algebra
(01'
)j e,
For
i s a Menger subalgebra of
e o n t a i n s a l l p r o j e c t i o n s and i s c l o s e d with r e -
.
F$
OF))
spect t o s
F f OA
(e i d e n t i t y ,
0 )
), l e t
f60:
equal t o
f
fvn
composition).
be t h e n-ary o p e r a t i o n which i s
up t o f i c t i v e v a r i a b l e s and which i s d e f i n e d a s
As a generalization of results in $4 and as a specializatfon of 5.2 we get:
5.5 Corollary.
Let
-
G g G 1 s SA, H s O A
(' )! and
FL
oP)
Then there exists I? relational algebra % =(A;Q)
7
-
)
QcRA or
p)
Q
S
R
:
'
)
,
(nrl).
with
-
resp.,
such that
_
L
I
G = Auk
%-
(=AutA~ automorphisms)
-
'
G = w-Aut &
(=w-AutAQ weak automorphlams)
H = ~ n&
d
(=EndAQ endomorphis~ns
)
om (gn,$ )
P =
if and only if
--
(=PolF)Q n-am polymorghi~ms
)
'I-
(i) G = FL1
(ii)
G
I
~
( v ) ~d
)
p)
F
is -.
a subgruyp-of
)on (and
- H i, g
-is-a Menger
(iv) F
G
SA),
On
FnSA
=
F t~'
:
0
(iii)
(-and
= Loc P
( 11)
eubsemigroup of OA
-
aubalgebra of OA
(cf. 5.4)
or
-
F = a-Loc F, reap.,
sea.
.
~ / - ~"')~
g
P
---.--r o o f by 5.2 (note Aut.Q = ( ~ o l ( ~ ) ~ ) ~, l(w-Aut
Pol Q 0
SF,
Q)Pn=Pol Q
(,End
fi
oA(I 1%
Clearly, because of (iv), the parts of (i) and
Remark.
--.--(iii) included in parentheses m e superfluous,
At the end of this paragraph we ask whether the wanted
relational algebra $=<A; Q) may be chosen to be of finite
similiarity type, i,e. whether
Q may be finite.
=
.
For f i n i t e A t h e answer i e n e a r l y t r i v i a l ; we have:
5.6
*--F-;rc- b
Proposition. d A clone
F -4 OA
of
i s the s e t
-
of f i n i t e
morphism~of 2 f i n i t e r e l a t i o n a l a l g e b r a (A;&)
iff there
--
Pr o of
ooo--o-
existe
by 5.2,
seJN
a l l polytype
s-Loc F = F.
such t h a t
note t h a t ~ 1 ' ) ' i s f i n i t e ( s i n c e A f i n i t e ) ~ .
Besides t h i s i n t e r n a l c h a r a c t e r i z a t i o n ( a second one can
be found i n p 6 / ~ a ~ l ( ~ 4.1.9)])
atz
t h e r e e x i s t s a l s o an ex-
t e r n a l c h a r a c t e r i z a t i o n f o r polymorphisms of f i n i t e r e l a t i o n a l a l g e b r a s of f i n i t e type by means of a c h a i n c o n d i t i o n i n
l
t h e l a t t i c e of a l l c l o n e s of o p e r a t i o n s ( c f . ~ i j / ~ a l ( $ ..3)]).
I n case of i n f i n i t e
A
we can prove only t h e following
weaker p r o p o s i t i o n :
Let
P r o p o s i t ion.
(i)
--
F g OA. For t h e c o n d i t i o n s
LO~{F)= F
&,
f o r every down-directed
m-
tern i ~ ~ ] i e 1o 3f l o c a l c l o s e d c l o n e s of o p e r a t i o n s
(i.e.,
F ~ = L o c ( F ~ ) t, / i , j t 1
f\
FinFj~Fk )
existence
of
Fi=F.
ieI'
There e x i ~ t sa f i n i t e s e t
Q
Fi = F i m p l i e s
i€I
1'21 such t h a t
(ii)
3 ke1:
nerated
the
n
-
-
g r R A (i.e.
2 f i n i t e subset
3 B: f i n i t e
v3/gt~
of f i n i t e l y
E-
:g = r F ( B ) )
such t h a t F = P o l Q .
-( i i ) ' There e x i s t s
F = Pol Q
.
a
-
finite set
Q gRA
-such t h a t
t h e following implications hold:
(i) + ( i i ) : + ( i i ) + ( i i i )
.
.
( i i ) : ( + ( i i i ) f o l l o w s from 5.2 s i n c e 3 a o I N : ~ s( 3
u ~ ~
iga
( i i ) i = + ( i'i )i s t r i v i a l .
-------
Proof.
(i)+(ii)l:
Let
I={Q
IQf
InvAF, Q f i n i t e ,
V ~ G 3Q B
p
and l e t
F -Pol Q
Q-
s e t of l o o a l c l o s e d ( I .11 a ) c l o n e s (2.5(i).) ( s i n c e
n F~ = r\
POI
Q =POI
u
Q'
2'
Qc 1
Qe I
QCI
e q u a t i o n ( + ) f o l l o w s from
( c f , 1.8)).
that
n PQ'F,
5.8 Remark,
------
POI
i.e.9
TF(B) )
F n FQ, =
Q
(4.1 )
Inv F =
LO~<F)=F. (The
{
~ o l i g ? = ~ oPl ~ ( B )BI f i n i t e , B S ~ )
Thue, by ( i ) , t h e r e e x i s t s a f i n i t e
Qa'
-
QGT. Then {FQ1 QII) i s a down-directed
for
FpuQf ) and
finite:
F=Pol
u Q where
QEL '
1s,
!Q
1's I such
l6 GiIQ\< 8,.
The s u f f i c i e n t c o n d i t i o n 5 . 7 ( i ) becomes a l s o
n e c e s s a r y f o r ( i i ) i f one t a k e e i n t o c o n s i d e r a t i o n i n f i n i tary o p e r a t i o n s . T h i s can be done analogeouely t o t h e r e s u l t
6.7 ahown i n t h e next paragraph. Therefore we w i l l n o t go
i n t o f u r t h e r d e t a i l s here.
For f i n i t e A, a l l c o n d i t i o n s i n 5.7 a r e e q u i v a l e n t ,
96
Concrete characteriaatfons 11.
(Characterization of relational systems via
universal algebras)
In 95 we asked for the characterization of related (operational) structures (like Autfr, pols) of relational algebras
%.
Now, we are interesteeted in the dual question
(which had been much more investigated in the literature):
How to characterize related rela~tionalstruc-bures (like
cons, 1nvR) of
universal
algebras & ?
This question includes the characterization of related operational systems (like Aut C .
,~
n&
d ) because operatione can
be considered as relation8 (cf. 1.3).
Theee problems will be covered (cf. §$7-114)by the following characterization problemr
6.1 Concrete
characterization problem:
..........................
------
Given a set A
and
Qi=Ei&R A (i I), does there exist a
universal algebra 8( =(A; F) (I? g OA) such that
Qi = Ei 0 PnvAF ?
Under which condition^ one can choone tho
o p n r n t l . o n n of
to be of bounded rank (i.e. of bounded arity) ?
The solution looks like follows:
P
6.2 Theorem
-
Let
-
(cf.[~z78(~bm.b)]).
u . There
Q =
Qi
i€
I
Qi&EirRA ( i s I ) and
e x i s t s a universal algebra ~ = ( A ; F >
with
-
-
( a ) FSOA o r
..."
( b ) F 5 0 A 1"
OA
(*)
, resp.,
(s~P)
such t h a t
--
Qi
i f and
--
Ei"InvAF
(~EI)!
-
only_ if
or
( a ) Qi = E i f i ~ O c [ ~ ] ( i C I )
( b ) Qi
-------
Proof.
= Ein a-LOC[Q] ( i h I ) , r e s p .
(.a)"+":
Since
by 3 . 8 ( i ) ,1 .I 1 a ' , LOC
f
QifInvF
LQ~SLOC [1nv
wehave
Q s I n v F and,
I?]= I n v F. Thus
E~ n LOC[Q]
EifiInv F = Q i S E i n L 0 c ~ ~ ] ( a i n c Q
e i s B i n Q) and w e a r e done.
F=Pol Q. Then (by 4.2a)
Bake
Ein I n v P o l Q = Ei
A
Qi
= Ei
n LOC
CQ]=
Inv F. Case (b) c a n be proved analogeous-
I
lye
I n a d d i t i o n t o 6.2 i t would be v e r y i n t e r e s t i n g t o have
a c o n d i t i o n f o r t h e f i n i t e n c o a of
(p.41
17).
For f i n i t e
6.3 P r o p o s i t i o n .
-
A
we o b t a i n :
clone
-A ---
--
r i a n t ~o f a f i n i t e a l g e b r a
exists
an
eclN
For i n f i n i t e
such t h a t
QsRA
------<A;F> of f i n i t e type i f f -there
Q 5 R A i s t h e s e t o f a l l inva-
s-LOC Q = Q.
I ( c f . 5.6)
A , we do n o t have such a f u l l answer.
C l e a r l y , j o t : TI: EI-LOC Q
for
F (open problem i n l ~ d n 7 2
tobeequal
= Q
i s sti.11 a n e c e s s a r y c o n d i t i o n
InvF
forafiniteset
F$OA(but
u n f o r t u n a t e l y no l o n g e r s u f f i c i e n t ) . I n t h e f o l l o w i n g propos i t i o n we g i v e a s u f f i c i e n t c o n d i t i o n by means of a c h a i n
condition.
6.4 P r o p o s i t i o n .
(i)
Let
LOC[Q]=
QGRA
Q
and
&,
c o n s i d e r t h e c o n d i t i o---n s
-
f o r every ----down-directed -s z -
tem { Q ~ J1413 of l o c a l -closed clones of r e l a t i o n s
-
7
-
(i.e.,vi/is~: Q ~ = L o c [ Q ~ ] , 1
~ kc1:
~ , ~ ~
Q Ii n Q j Qk)
fhi= Q
if21
1's I
(ii)
-- t h e existence of
-lies
such t h a t
f i n i t e subset
n =~= Q .;
i&I1
Q=InvF
for 2
finite s e t FgOA;
Then t h e f o l l o w i n g i m p l i c a t i o ns hold:
7
=+( i i ) =+ (iii).
A, a l l conditions a r e e q u i i a-lent.
(i)
For f i n i t e
-
-------
proof.
~ F I 5Pol
F
{Q~IFEI)
-.
I k r e m a i n s ( c f . 6.2) t o prove ( i ) + ( i i ) .
Q and
P finite)
and
Qp=Inv F
I=
f o r FEI. Then
i a a down-directed s e t of l o c a l c l o s e d c l o n e s and
nQF=nInv
F =Inv U F =Inv p o i
PtI
FeI
FEI
by ( i ) , t h e r e e x i s t s a f i n i t e s e t
n
Qp=Q,i.ee,
FtI1
6.5
-
Let
Q i 4 * 2 ) ~ 0 c k ~ 3 T= ~~U. S ,
I
-
I
such t h a t
Q=I~V
F U
with 1 U t ~ ( w o
Ie
F6.1 l
FCI
U n f o r t u n a t e l y t h e a u t h o r w a s n o t a b l e tcl f i n d " i n n e r f t
conditions f o r
Q
which a r e e q u i v a l e n t t o 6 , 4 ( i i ) i n c a s e
142 8,. Condition 6 . 4 ( i ) l o o k s on
Q
only "from t h e o u t s i d e l I
because one h a s t o c o n s i d e r t h e l a t t i c e o f a l l c l o n e s on
A.
--A.
/*-A-
c
A / A \ \
-
-
b. - - - - a m - - -
- 2
-...e.eA-*--L
sideration to be less than or equal to /A(?%.
weedefine
CJoO
---..¶2A.2--
For Q s R A
.
:= lnv@~ol Q
90 give some aense to the theorem below we mention (but will
8-
no4 go into details) that
M C [~'l,(analo~eousl~to
can be characterized as
Inv Pol Q = LOG [Q]
)
where ,]Q[
is
the clone of relations(of arity L, \A1 )' generated by Q. The
definition of the clone
tion of 3.5 ([Q]~
/h],
is e quite natural generaliza-
is the closure with respect to 3.2(RO),
(R1 ),(R2) and (R3))s.
6.6 Lemma.
-
<F)=
F5OA,
6.7 Theorem.
-
For Q g R A ,
Pol 1nve0F (cf.~kr/Poi~[~os72~
the following conditions are
-
equivalent.(assume lh128$) :
(*)
Q=Loc[Q?~
for eve-
A
and
-
-
down-direct ed
- . system {Q~Jic13 of
local closed clones (cf. 6.4(1))
f l QY =
ie1
implies the existence of-a finite set
such that Alpi
= Q
-i€I
(st) There exists a finite
.
set P S O A
-
I1gI
with Q = InvAF.
(*)+(*+): Let I=+( F g P o l Q , F
.---.--
f i n i t e ) and
Proof.
QF'Inv F. Then
%=Loc[Q~]
and { Q ~ J F ~ I )i s
Pol Znv P o l Q = P o l Q, i.e.,
Pol Q = u p 0 1 Inv F. Therefore
r ) 'QF= Q f o r a f i n i t e I
I, i
,
FLI
= 1nv
F where (
F 14g0.
FCI I
F~I'
(,*)+(r): Assume Q = I ~ v F F, f i n i t e . Let
By ( x ) ,
u
down-directed
FIFGI~~
{ S Inv
Pol Q = U ~ / F ~ I VfPol
system. We have
R
Q =/)
Inv F
FtI'
u
~ Q Y = Q and
@
3x1
We d i v i d e t h e proof of t h e e x i s t e n c e of a f i n i t e
Qi=LOCrQi].
I ' i n t o s i x p a r t s a)-f), We have:
Q?
a ) Pol
=Pol
Qi
s i n o e Pol p
:
C)
(@=L(F},
Pol lnvdF =(
u
Pol Q? , i n f a c t r
i c1
= 1 n v W ( u ~ o l )=Inv00'"F
N
b ) 1ngoY = QoO where F: =
n~nv??o
: Ql
=Pol ~ n v * ~ o Qii6*6@01
l
41).
since
.
oON
Inv F = b ) ~ m q
=- ~
= ~#
nP
~~Inv
~
o llF L l n t e
1 n v m < ~ ) ,t h e r e f o r e Pol I#$Z
d ) ~ I ' C I , 1' f i n i t e : < F > s ( , ~ Pol
implies
exiets
i&IL
QT),
pol 1nvW(3)
there
irI1
~ rA)Inv
n
n
.
since: By c), F ~ ( ? >
3 ? 1 s $ , ? f i n i t e : F s < F 1 > . By d e f i n i t i o n of F,
a f i n i t e 1's I such t h a t ( C/ Pol QT)~(F')<F).
Q~ , since: Q $ ) I ~ ~ ( C PO^
/
i€I1
itI1
=a)r)1nv p o l Q~ =
Q ~ .W
e a r e done.
iEI'
i€I1
f) qa
Q ~ = ~= Q T
=
ikI1
I
PO^ ;Q
$7
Concrete characterizations
---.
--- 111.
-
,-
- -
7.1 Characterization
~roblem:
-----------.---------Let
A be a set,
formation semigroup, L
let
a permutation group, H S O A(' ) a trans-
G&SA
C be a subset of
a subset of 2A (power set of A) and
g ( ~(set
) of all
equivalence relations
on A ) . Does there exist a universal algebra &=(A;
)
3 5 OA arbitraxy
(2) F~
)
b
)?I
where
("general caseu)
OiB)'for some B E I N ("bounded casett)
finite
("finite casetr),
such t h a t
G = A U (41
~ (automorph%sm group) and/or
H = ~ n d (endomorphism monoid) and/or
L = Subs (aubalgebra lattice) and/or
C = ~ o n a(congruence lattice) ?
(i) Because of 1.6d) we need not distinguish betRemarks:
----.-ween w-Au* and Aut.
(ii) Without loss of generality one can assume at once
L
and
C to be algebraic sublattices.
Theorem 6.2 provides an answer for case
(A) and
-
(B)
-
while case (C) can be treated with 6.4 and 6.7.
Soma of the above problems
T.1 have a well-known answer,
some others were still open (in particular the simultaneous
characterization of G,H,L,C). The known answers sometimes
are better than those given by 6.2 because the use of clones
of relations is avoided or reduced to simpler closure properties. In general however, we think there is little hope to
find conditions for G, H, L and C which are not based on
closure properties of the clone of relations generated by
GeuH'uLvC.
We have:
7.2 Theorem. Let
y
-
-
G, H, L and C --as in 7.1
Q = G'u He
L y C. Then there exists 2 universal algebra O={A;F>
A)
-
F G O A op
(B)
.-
F ~ o ~( s ~ N))reap.,
such that
7-
, H = End a
L = sub tk , c = Con &
G = Aut 8(
if and only if
-(_A)
G 7 = S A n H,
H* = (o~))*~Loc[Q~,
L = LOC [Q]
c=
)
,
or
~ ( An
I LOC [Q]
(B) O = S A " H
H' = (OA I)'n e-LOC[Q]
,
L = S-LOC [Q] (1
c=
f(~)s-LOC[Q]
r\
resp.
B (6.2)
with
Another application of 6 - 2 provides the characterization
of bicentralizers of universal algebras (F=Pol Pol F f is
------------------icentralizer
called Dhe b
Proposition.
For
of Ft_cOg):
F g OA, F is-a bicentralizer iff
P* = og* ~ L O C [ F * ] .
-------
P r o o f . by 6.2, note (Pol Pol Ft)*=OA*nInvPol F v .
-------
for finite
Remarks: 7.3 can be found also in [~z78(~hm,13)];
A this is a result of A,V, Kusnecov (cf,
[
~
a
]
$
z
'
l
8
J
)
.
Sufficient conditions for F to be a bicentralizer were also
given.in
]a@,
Bicentralizers in O3 are described in.]ina~[
-
In the next paragraphs we list (most of) the problems
in 7.1 and (some of) their solutions adjoining some (but
surely not all) references. The propaxitions are marked with
( A ) , (2)or (g)
whenever case 7,1(A),(B)
or
(c) is treated, The notations of
this paragraph will also be used in the following ones.
It is worthy to note that all the characterization problems above have a non-tfvial solution, i,e. not all structures under consideration are related to some universal algebra. We do not mention this fact explicitely every time
but it follows directly from the corresponding characterization theoreme which allow the construction of counterexamples.
$8
Concrete c h a r a c t e r i z a t i o n of Aut
a
The f u l l answer t o t h i s problem f i r s t l y w a s given by B.
.
J6nsson i n fi6n681 (cf n 6 n 7 2 G2.4.3
(A)=
~heorem([Jbn68])~.
3a
I],
: G = Aut fi
namely:
+>
G = SAn L O C < G > ~ 6
A
-i s t h e automorphism
grou2 -of some a l g e b r a -i f f t h e following c o n d i t i o n holds:
group
pemuta tions
(on A )
i f f o r every f i n i t e s u b s e t B
For every hrSA , e x i ~ t sa member of G t h a t a g r e e s with h on
7
------
Remark:
of
-A
tihere
-
heG.
7
B, ahen
(X can be chosen a s a simple a l g e b r a ( c f . li3.5).
The f o l l o w i n g answer f o l l o w s from 6.2:
--
( ~ j 8 . 2P r o p o s i t i o n .
3 a : F = A U ~ Q <=>
Ge=SA ~MC[G*]..
Note t h a t t h e r e s u l t 8.1 i s much b e t t e r t h a n 8.2 because
t h e l o c a l c l o s u r e of t h e clone of o p e r a t i o n s generated by
G
i s much l e s s complicated than t h e l o c a l c l o a u r e of t h e clone
of r e l a t i o n s generated by
G'
. But t h i s c a s e s h a l l s e r v e u s
as an example t o show how t o g e t completely new proofs w i t h i n
t h e framework of o u r General G a l o i s theory; we s h a l l s e e how
t o work w i t h clonee o f r e l a t i o n s (because i n o t h e r c a m s only
t h i e method works). Therefore we g i v e t h e
Proof
of 8.1
w i n g 8.2:
-o--o----------
and G=SAn Loc G (4.6).
C l e a r l y , G=Auta i m p l i e s G = < G > ~
ll+tl:
A
We show
tt+tl:
SL n
~ o ~ [ =~(SA
* ]n LOC<G)
because t h e n 8.1
)I@
S~
immediately f o l l o w s from 8.2, F i r s t l y ,
(S,ALOC(G>
A
= Si n
)*
~ ~ ~ by
[ ~
4.7.* ]
Secondly we show t h e opposite i n c l u s i o n . Since (G)
we can assume
G = < G ) ~ ~ . Let
' g [G']
SA
f m tS ~ ~ L O C CWG
e must
~ . show
By d e f i n i t i o n 1.9, f o r each f i n i t e subset
-B gf'
there e x i s t s
a g e n e r a l s u p e r p o s i t i o n Q E ~ G ~such
]
that
-B S G f
f'.
Note t h a t t h e r e i e a 1-1 correspondence between 3 and f i n i t e
BcA
by
B ={r
(
sf'$
Since 6e[G03 t h e r e e x i s t
-2 +DL
f o r s u i t a b l e Ti:
and we have
gicG
B = ( ~ J B ).'
( i c 1 ) such t h a t
(d o r d i n a l ) , of.
3.2(R4).
Consider ( t h e q u a n t i f i e r f r e e p a r t o f ) t h i s formula a s a
l a b l e d graph w i t h v e r t e x a e t
V ={a. 1 j
J
~ d ) ,and f o r
(7ri(0)1,7ri(l)) = ( t , t l ) as t a k e a n arrow from
with l a b e l
gi
(ipI).
at
to
at,
We d i s t i n g u i s h two a a a e s (Remark: t h e
used arguments t u r n out t o be here t h e same a s discussed i n
f i a / ~ t 7 7 c( p . 225 )]
Caeg-lLl:
i n t h e language of equations i n semigroups):
The v e r t i c e s a.
aO=atO,atl
exist distinct vertices
t h e r e i e a n edge from at
l a b e l gi ( j r y ) . Put
t o at
,...,a
o r from at
=al
i n t h e l a t t e r one.
3
there
B U C ~t h a t .
t o at
3
3 +1
j+I
gi :=gij i n t h e former case and
3
gi
and a, a r e connected, i.e.
with
3
gi :=
j
Then we get for
i.e.,
6 ~ g where
'
-
:
g = g ! gi
Since B G 6 we have
...
lo I
-Bgg'.
But
=G.
€<G>~
gin-^
A
this implies c g ( ~ ) * 8
=,
i.e., ~ J =B f ( ~(and gcG).
Case Liil:
----.
I
The vertices eo and al a r e not connected. Let
('~c,y),(xl,yl)
.
be distinct elements of B $ c7 By the diecon-
nectednees of .
a and a,
, the
element (x,yt) also satisfies
the above formula for 6e[~*l, i.e. fx,yl)€& in contradiction
-
to 6 -~ f * ( ( f € Stherefore
~
f(x)=yfyl-f(xl)). Clearly, B must
have at least two elements what we can assume without loss
of generality.
Thus only case (i) occurs: For all finite B p- c A there is
a gaG with
f(~=gl~,i.e.f~LocG. H
-...--
If one permits to uee algebras with infinitary opeRemark:
rations then every permutation group G is the automorphismgroup of an algebra (cf.[Ar/~chmJ).
Let us consider now1the "bounded casetl(B)(cf.
7.1). This
case was treated by E. P&onka
too:
Defining (for G$SA)
[PSI
and
B e J6nsson fi6n722,
vf
8.3
-
: ~ J =B~ I B
c ~ ( B ) : = { ~ € A ( ,@G
BG
- A,
f(a)=g(a)] f o r
t h e y o b t a i n e d t h e following theorems:
(B)=
u ).
~heorem([~&],[~bn72 (2.4.1
--
~ = < A ; F ) with
~ ~ 0 P ) ( a such
~ 2 t)h a t
-a permutation jqroup and
all B5 A --with a t most
t h a t agrees w
ith f on
-
There e x i s t s
f € S A belongs
a
an a l g e b r a
-
G:=Aut8 i f f
5
G
is
-
G , whenever f o r
-
elements t h e r e e x i s t s a
goG
CG(B).
3 F ~ o :G~ = Aut
)
F
t(fosA(\/ac~: CG (-la)) # la) or C G ( {f ( a ) f
( B ) u ~heorem([Jbn72(2.4.4)]).
G = <G&
A
+
#
ff (a)]
%3peG : g J c G ( ~= )f ( ~ G ~ f a j )
Let u s compare t h i s with t h e r e s u l t which f o l l o w s d i r e c t l y
from 6.2 f o r a r b i t r a r y
--
There e x i a t s an -a s e b r a
( ~ j 8 . 6P r o p o s i t i o n .
F
~
selN:
-.-
--
O -such
~ ) that
--.,a.
G = A U ~ Uiff
O=<A;F)
G* = ~ ; A S - L O C [ G ~ .
--
Wlth
a
We s e e t h a t i t i s p o s s i b l e ( c f . 8.4 and 8.6) t o d e s c r i b e
t h e elements of
s-LOC[G
which
~
a r e permutations ( i . e .
ele-
ments of S d ) i n terms of permutations and very s p e c i a l r e l a tions
CG(I3). O f c o u r s e , one can deduce 8 . 4 from 8.6 i n t h e
same manner a s 8.1
from 8.2.
The s t r u c t u r e of C G ( B ) can be d e s c r i b e d a l s o a s f o l l o w s
(cf. 8.3):
8.7 P r o p o s i t i o n .
-
For
B = {bl
,...,bn] 5 A
and G < SA -we have:
-
...,bn)l ~ B P O ~ ( ~=) Gr S
cG(~)={h(b,
l
-------
Proof.
POI
Clearly, f o r
f ,geG
and flB =
=
=
prove t h a t t h e r e i s an
("1
hrOA
We show
h(bl
,
aeCG(B). It remains t o
h ~ ~ o l ( " ) Gsuch t h a t
a = h(bl
,...,bn) .
a s follows:
h(fb,
,..., f b n ) :=
h(xl,...,x
n ):=xl
f (a)
for
feG
(9,
for
-0
and
,Xn
) Q~ := A" \
{(fbl
Then, f o r
~ p .(11 ).
...,bn) ( h e ~ o l ( " ) G ) .
glB we have f ( h ( b l ,..., b n ) ) =
g ( h ( b l ,..., b n ) ) , i . e . , a l l
..., f b n ) h(gbl ,...,gbn)
h ( b l , ...,b,) belong t o C G ( B ) o Let
,
Define
(B)
( n ) ~
The r i g h t p a r t f o l l o w s from 2.4 and 4.3.
t h a t CG(B) contains e x a c t l y a l l
h(fbl
L
,...,fb,)lf€G).
geG, we g e t
g(h(fb1 a..,
,fbn)
= g ( f ( a ) ) = h ( g ( f b 1 ),.-, g ( f b n )
g ( h ( x l ,*w,x
n 1) = g ( x l ) = h(gxl ,--,gxn);
or
( n o t e t h a t (xl
,...,xn)t!p#
c o n s e q u e n t l y , h ~ P o l ( " ) G and
A cornparision of 8.1
,...,gxn)€p because
a = h(bl ,...,bn). 1
(gxl
and 8.2 w i t h 8.4 and 8.6 l e a d s t o t h e
q u e s t i o n which r o l e p l a y s t h e c o n d i t i o n
case
(2). The
G i s a group),
G = s-LOC<G> f o r
SA
r e s u l t coincides (for ~€37)
w i t h Lemma 2.4.2
i n [~bn32/:
--
( ~ ) 8 . 8F r o p o s i t i o n .
For
-
GCSA, S E B , c o n s i d e r t h e c o n d i t i o n s
(i) G , = s , n s - ~ o c ( ~ >
SA
G = AutAF ( o r e q u i v a l e n t l y ,
(ii) 3 F _C 0p)':
.
(iii)
G = SAq (s+I )-LOC<G>~
A
Then t h e f o l l o w i n g i m p l i c a t i o n s hold:
-(i)
-------
Proof.
+ ( i i l +( i i i ) .
G=SAns-LOC(G) ( c f . 1 . 9 ) t h e n t h e c o n d i t i o n
SA
i n 8.4 i s f u l f i l l e d s i n c e
~ J c , ( B ) Jc,(B)
=~
i m p l i e s g B=f
consequently G = Aut F f o r some
F s
- OA
by 8.4.
If
1
Now, i f
G=Aut F = S A f i P o l F *( c f . l . 6 d ) ,
we have
G = S A n(s+l)-Loc G ( s i n c e
iB,
F ~ O ! ~ ) , t h e n , by 5.2,
F 0 s R (As + l ) ) *
I n 8.8, t h e i n v e r s e i m p l i c a t i o n s do not hold i n g e n e r a l
(aa shown i n f i 6 n 7 2 ( ~ 37)J).
.
n i c e r e s u l t of M.
The next theorem w i l l show t h e
Gould t h a t t h e s o l u t i o n of t h e l1bounded
-
c a s e 1 I ( ~ provides
)
a t t h e same time t h e s o l u t i o n of t h e "fi-
-
n i t e caselt(C) ( c f .7.1).
This r e s u l t a l s o shows t h a t t h e con-
d i t i o n s i n 8.8 become equivalent a f t e r q u a n t i f y i n g a .
(C)8.9 ~heorem([Do72a]).
For Gc SA, t h e following
conditions
a r e equivalent:
(i)
3
(ii)
~ S G I N ~ F ~G O
= A
~ U~ ~ ) F :;
3
3
(iii)
(iv)
-------
Pr o o f.
finite
f€oA :
sea :
F E O A . * G=Aut F ;
G =A
{fi
U ~
;
.
G = S ~ ~ ~ - L O C ( G > ~
Clearly ( i i i ) + i ) + ( i i )
A
(8*8&
(iv)
.
For ( i ) + ( i i i )
( e a s y t o do) and ( i v ) = + ( i ) ( c r u c i a l p o i n t of t h e p r o o f ) we
r e f e r t o [~072a(pp. 1066,1067)].
(1)
8.10 Remarks.
-------
a) Loc G (or a-Loc G resp.) is the least group which contains a permutation group G<SA
and which is at the same
time the automorphism group of an algebra (or an algebra
with operations of rank 5 a, resp., sclN). A similiar observation fails for algebras with finitely many operations: If
--
is an infinite chain than - by 8.9 - there does not exist a
least automorphism group of an algebra fi = {A;F) with finite
F and G -S A U ~ ~ .
Example :
Let A =u{An1 ntJN1 be the union of disjoint sets An with
lAnJ=n. Let G C SA be the group consisting of all permutais the
tions fsSA such that (f(A2€ 8, (for n c ; l where
alternating group on An).
Because (n-2)-Loc
= Sn > % but
(n-1 )-Loc Lln = %
one easily proves that
SAns-Loc G >
6 SA n(s~-1)-Loc G (for ~ 1
-2). 1
-
7
--
-
an
%
b) For the concrete characterization of the (lattice of all)
Y
automorphismgroups
of all subalgebras (of a given universal
algebra) we refer to ]OK[
(for the abstract version see[Fr/~o.
The semigroup of all local automorphisms is investigated and
r] .]57zs[,
characterized in n
58
$9
Concrete characterization of Enda
3 U=(A;F):
H = Enda
?
M. Armbmst and J. Schmidt showed in fkr/~chm_7that every
(abstract) monoid is isomorphic to the endomorphism semigroup
of some qniversal algebra. They observed also that the concrete characterization problem for transformation semigroups
has a non-trivial solution (i.e. there are monoids which are
not equal to the endomorphism semigroup of an algebra).
Obviously, the existence of an fl with H = 6nd&
is
equivalent to the condition H =End Pol H. In [sa/~t77c]
the
set E n d P o l H ( f o r ~ ~ ~ yis
) )
called the a
--l g--------ebraic
c
l o s u r e of H .
---------As N. Sauer and M.G. Stone pointed out in [sa/~t77c],
the
determination of the algebraic closure of semigroups is relaL
j (p.25)]
and
ted to broad questions posed by E.S. Ljapin [
S. Ulam fUl
32)]
regarding the determination of algebraic
structures from given endomorphisms.
By a result of J. Sichler [Si](for
finite A, IA115) the algebraic closure of H contains all maps in OA
(' ) if H- contains anything more than constant;.mapsand all of the permutations f€SA* If H consists dlocally invertible and constant maps only, a necessary and sufficient condition for H
to be algebraic (i.e. H=End Pol H) is to be found in Bt75J
(cf.[st69]),
where results of W.A. Lampe and G. Grztzer
,
]
8
6
a
~
[
(
f~r/La68])
had been generalized.
N. Sauer and M.G.
Stone gave a characterization of the
algebraic closure of H for H = If) (f60F ) ) in [Sa/~t77b;l
and for arbitrary H C:0 ) in [~a/st77c]
in terms of "equational conditionsw. The same question (which was posed explicitely as problem 3 in L?;r(p.77)])
is treated by L. Szab6 in
fiz78(Thm.15)]
who gave a characterization by means of formula schemes.
Our approach provides a c h a r a c t e r i z a t i o n theorem i n terms of
c l o n e s of r e l a t i o n s :
From 6.2 we conclude d i r e c t l y :
-
(A)9.1
- - ~ h e o r e m ( c f . [ S z 7 8 ( ~ h m . l 5 ~ . For
an a l g e b r a
-
with
@=<A;F)
For
~B)9.2Theorern.
with
(i)
iff
-
H=Endcf
there e x i s t s
He = o ~ ) ~ ~ L o c ~ * ] . I
H 5 0 r 1 , s a A , there e x i s t s ~ a l g e b r a
and H = ~ n d &iff
-
P$OA
remark^.
-------
&9
HSO:),
-
The a l g e b r a i c c l o s u r e of
5
H
OF
)
H*=O~)~,S-LOC[H~.B
i s t h e l e a s t endo-
morphism monoid c o n t a i n i n g H. Thus we have (by 9.1
For
-- -
H z ( ) A ) 9 -,t h e s e t of a l l
,
(-o r ~ ' G S - L O C ~ H ~r e s p . )
----
is the
--
fc0;
, 9.2)
:
~ * ~ L o c [ H ~
) with
l e a s t semigroup which c o n t a i n s
.
.
1
1
1
-
of an a l g e b ~( o r an
H and which i s t h e endomorphism monoid -
--
-
a l g e b r a -w
. i t h o p e r a t i o n s of -rank
4
(ii)
End
The o p e r a t o r s
Pol
and
Z
s, resp.).
d e f i n e a Galois connec-
t i o n between s u b s e t a of OA ) and s u b s e t s of On.
The Galois
c l o s e d s e t s with r e s p e c t t o t h i s Galois connection were characterized
i n [ S a / ~ t 7 8 ] ( F = ~ o l End F) and [ ~ a / ~ t 7 7 c ] ( ~= End
Pol H, c f . 9.1).
Because
Galoie connection
Inv-End
Pol-End
i s t h e r e s t r i c t i o n of t h e
( i . e. 1nv-pol (l ) ) t o o p e r a t i o n s
o n l y , t h e r e s u l t i n [ ~ a / ~ t 7 8 ( ~ h m17
. l can be considered a s a
s p e c i a l c a s e of theorem 4,2(b]e(or 6.2) f o r
s = 1 . In fact,
we have:
P r o p o s i t i o n : F = P o l End F
F* = O i n I n v Pol (1 I,?
.
@ F = o ~ ~ ~ - L o cD
[ F ~
Note t h a t
1 -LOC [F']
i s easy t o d e s c r i b e : Take t h e clone of
r e l a t i o n s g e n e r a t e d by F'
and t h e n t h e c l o s u r e w i t h r e s p e c t
t o a r b i t r a r y u n i o n s ( c f . 1 .1:3).
( i i i ) : The c o n d i t i o n f o r H i n 9.1 cannot be r e p l a c e d by
(1 )! (~hm.9.6 ( o r Thm.1
OA
examples).
H = LOC(H)
(2)9,4
i n [St7511 ~ r o v i d e ec o u n t e r -
-
F o r t h e f i n i t e c a s e (C) o f t h e c h a r a c t e r i z a t i o n problem
we do n o t have such a good s o l u t i o n as theorem 8.9 f o r g r o u p s
From 6.7 we g e t :
3 finite
F r H=EndAF
H
=
)H
O
and
s a t i s f i e s 6.7(*).
Q = LOC[H*]
But t h i s c o n d i t i o n i s n o t v e r y s a t i s f a c t o r i l y ,
A t t h e end of t h i s paragraph we p r e s e n t t h e a - l o c a l v e r -
~ t o n e f ~ t 7 5 ] ( w h i c hi s an e x t e n s i o n
s i o n o f a r e s u l t o f M.G.
o f p r o p o s i t i o n 8.8 t o c e r t a i n monoids), A c a r e f u l examinat i o n of t h e proof o f theorem 1 i n [st751
shows t h a t , i n f a c t ,
t h e r e w a s proven theorem 9.6 below, t o o . However we s h a l l
p r e s e n t a n o t h e r proof based on t h e g e n e r a l c h a r a c t e r i z a t i o n
theorem 9.2
( o r 6.2).
9.5 D e f i n i t i o n s .
( ~ e m if
)
C a l l a monoid
s - l o c a l l y
------me----
--------- i--------------n v e r t i b l e )
l o c a l l y
respectively), for a l l
f ( a i ) = g ( b i ) ( i s n-)
such t h a t
#
-----s - a 1 g--------e b r a . i ~
~ ~ = ( A ; F >~: ~ 0 : and
~ ) M=End@
~ -~ 0 r ) s ai i ds t o be
f,gtE,
IsOA(1
h(ai) = bi
-
A rnonoid
--------------i n v e r t i b l e
if for all
(ai)iEny (biIien
.
-e
n =4 s ( o r
n~lN,
and f o r a l l
i m p l i e s t h e e x i o t e n c e o f an
.
(ieg)
(or
heE
I n an s - l o c a l l y i n v e r t i b l e monoid c s z 2 ) each map i s i n j e c t i v
( b u t n o t n e c e s s a r i l y s u r j e c t i v ) . Let
ca: x
function
For
ca
be t h e c o n s t a n t
a.
I-+
M 5 OA(' ) we d e f i n e :
X(M)<:=(C~
1 VbeA,
Let
( 2 ) sTheorem(cf .8.8).
bfa,
3
f ,geM : f ( a ) = g ( a ) & f (b)fg(b)].
--
s€lN and l e t
-
M =EuK
n o i d where
--
E f O? ) i s s - l o c a l l y i n v e r t i b l e and
of constant
-
maps on A. Then t h e i m p l i c a t i o n s
-
K
-be-a-mo22
set
-
7-
hold f o r t h e
---
following conditionsr
and
-
is s-locally
-
closed
i s -s - a l g e b r a i c , -i.e. 3 ~ ~ 0 1 M=EndAF,
~ ) :
( i i i ) ae(M) M and l'il i s (s+l ) - l o c a l l y c l o s e d
(i . e . M = ( s + l )-Loc M).
(ii)
9.7 Remarks.
-------
stED
a ) From 9.6 we g e t immediately t h e f o l l o w i n g
is locally invertible t h e n M -i s s-al-f o r some SEIN iff M i s s f - l o c a l l y c l o s e d --for some
Proposition.
gebraic
M
&
If
E
ae(M)sM.
I
b ) T h i s p r o p o s i t i o n (and 9.6) show ( i n comparision w i t h 9.2)
t h a t t h e r e s t r i c t i o n t o c e r t a i n monoids b1 ( o f a r e l a t i v e l y
amall c l a s s ) l e a d s t o an improvement of t h e c o n d i t i o n s f o r
c h a r a c t e r i z i n g s - a l g e b r a i c i t y , namely t h e involved s - l o c a l
closure of
can be r e p l a c e d by t h e s i m p l e r s - l o c a l c l o s u r e
of
1 ) ( = M s i n c e M i s supposed h e r e t o be a monoid).
[MI
d
-------
Proof
o f 9'6
( u s i n g 9'2):
We have ( i i ) + ( i i i )
therefore
by 5.2;
... ,a)=*,
h(a,
would imply
=g(h(a,
M=EndAF i m p l i e s
M=(s+l )-Loc M
~ E F ( " ) we have
f a
since
f(b)=f(h(a,
caea(M) and
carzld, because
h(a
We w i l l show M e = :0
Mec
- (0:
Obviously,
...,a ) ) = h ( f a , . ..,f a ) = h ( g a , ...,g
. Then,
t h e r e e x i s t s an
we proceed as i n t h e proof o f
) *n s-LOC
[M*]
, then
) ) '~S-LOC[M']
.
f€0Y )
B Simply
CM*]
and
f 'c
such t h a t
(cf. 1 e 9 ) .
6 $ f'
We v d l l f i n d an
we a r e done
Bg A , I B I s s , ~ = { ( x , f( x ) ) l x e ~ f = ( f ( ~ ) ' ,
for
Qt
a)
f ( b ) f g ( b ) (where f ,g
To show t h e o p p o s i t e i n c l u s i o n , l e t
B-LOC[N']
,..., a ) = b
x(M), c f . 9.5).
F o r t h e proof o f ( i ) + ( i i )
by 9.2.
moreover, f o r
., , a ) ) = g ( b ) i n c o n t r a d i c t i o n t o
a r e as i n t h e d e f i n i t i o n of
8.1 r
i.e.
M=EndAFe and
such t h a t
grh1
g *2
x, o b v i o u s l y
this w i l l
fIB = glB, i . e . ( s i n c e B was chosen a r b i t r a r i l y )
fcs-Loc M = M (by ( i ) )
, consequently
hl* -2
Oil
) ' 0 a-LOC
Me
and t h e proof w i l l be f i n i s h e d .
HOW t o f i n d now t h e
geM ? S i n c e
& c [ ~ * ] there exist
gieM
( i € I ) such t h a t
@ ](aj )jcor\2 : ( 9 r i ( 0 ) , a ~ ~ () €l p) i (-1)
(ao' a1
f o r s u i t a b l e Ti:
2
+d
-
(d
ordinal, cf. 3.2(R4)).
We c o n s i d e r t h i s formula as a l a b l e d graph w i t h v e r t e x s e t
V ={aj)j t g } such t h a t
some IcI, "zi(1)= "j
a.EV
J
and
w i l l get the label
gieK
cb i f , f o r
i s t h e c o n s t a n t f u n c t i o n cb
(we c a n assume b # (8, t h e r e f o r e t h i s l a b e l i n g i s c o n s i s t e n t ) ;
moreover, f o r
(edge) from
(ari
at
to
(o) ,ani (1 ) )
at,
= (at ,at, )
with l a b e l
gi
we t a k e an arrow
whenever
gicE
(more t h a n one edge between two p o i n t s i s not excluded).
\Ye d i s t i n g u i s two cases.
------
Caae 1 :
The v e r t i c e s
t h e r e e x i s t s a constant
i . e e , ~ ~ c d aoi n, c e
d€A
al
a r e not connected. Then
such t h a t
(x,y)cs
( x , y ) , ( x l , y l ) ~ &imply
connectedness, t h e r e f o r e
We a r e going t o show
Subcase 1,a:
----------
and
a.
because of
y=yl
+y = d ,
(x,yl)~@bd
yis-
@sfe(~Eo:
) ).
cd€M.
Bhere i s a v e r t e x
ai
which i s connected w i t h
al and l a b e l e d with a c o n s t a n t c b , i.e. ne have t h e f o l l o wing s i t u a t i o n ( t h e l a b e l s a r e put i n p a r e n t h e s i s ) :
Now, we llmovell t h e c o n s t a n t l a b e l from
ai t o t h e v e r t e x
al by t h e follovring i n d u c t i o n s t e p s (which do not change t h e
p r o p e r t y of (ao ,al ) belonging t o
(r
, i. e.
we change t h e above
formula f o r 0- w i t h o u t changing t h e r e l a t i o n & d e f i n e d by
t h i s formula):
@-
"b)
(g')
i a equivalent t o
(e')
(
~
~
1
@
) ( c~b )
Since E i s s - l o c a l l y i n v e r t i b l e , g l ( a t ) = a t l = e ( a t l ) ( e i d e n t i t y ,
g'eE) i m p l i e s t h e e x i s t e n c e of an
h€E
such t h a t
a t = h ( a t ,)
and we have:
i s equivalent t o
at
( ~ b1*a-' (g')
at '
"t '
at
or to
(
~
( cb )
~( h )1 @) ( c b~)
at
with
at '
b 1 = h(b).
I n b o t h s i t u a t i o n s , cbl€LI ( s i n c e . g l , h , c b ~ B ) . A f t e r n s t e p s
we g e t a l a b e l
cd ,EM
al =d I . We have
d=d
Subcase 1 b :
----------
f o r t h e v e r t e x al
, i.e.
(ao,al
)€a=+
s i n c e (ao , a l )ecr L+ al = d o Thus 05 c d e M *
.
There i s no v e r t e x which i s connected w i t h al
and l a b e l e d w i t h a c o n s t a n t . Ne prove
( a O , d ) e r but
( a o , d l ))o. f o r
cdfac(Fil). We have
d # d l (a0eB). We i n t e r p r e t
t h i s s t a t e m e n t i n t h e graph V which r e p r e s e n t s o u r formula e,
and g e t :
If we l a b e l al w i t h ( c d ) o r ( c d l ) , r e s p e c t i v e l y , and i f we
l a b e l ( i n a l l p o s s i b l e ways) a l l v e r t i c e s connected w i t h al
c o n s e c u t i v e l y by t h e above i n d u c t i o n s t e p s t h e n a l l t h e o b t a i ned l a b e l s ( f o r v e r t i c e s ) w i l l be c o n s i s t e n t (compatibxe) o r
i n c o n s i s t e n t , r e s p e c t i v e l y . That i s , t h e r e e x i s t two p a t h e
from al t o a v e r t e x , s a y ai,
(by t h e above e q u i v a l e n t
t r a n s f o r m a t i o n s we c a n
assume t h a t a l l arrows
have t h e same d i r e c t i o n
from al t o ai)
a
such t h a t
h=hlh2
h ( d ) = h t ( d ) but
... hnOM
condition f o r
with
and
cd
h ( d l ) # h l ( d t ) where
h1=hjh$
... GeM. T h i s i s e x a c t l y t h e
t o be an element of *(M),
i.e.
B C C ~
c d E x ( M ) ~ M (by ( i ) ) .
3
Summarizing c a s e 1 we g e t
Case
------2:
The v e r t i c e s a.
gcM: B s g * ( t h e wanted r e s u l t ) .
and al a r e connected. If
t h e n we can assume t h a t t h e r e i s a p a t h from a.
followe (with
ho,
(hO)
...,hn-l aE)
(hl)
- ...*- (hpl
:
,,
,I
a ~ = a t o at
9e
at 2
1
__ED
at n-1
a tn =al
aoeB,
t o al
as
T h i s i s c o r r e c t because
(gl1
.t.-----.
at
- by
t h e s - l o c a l l y i n v e r t i b i l i t y of E
i s equivalent t o
at '
(h)'
-0
"t
f o r some h6E i f a t most
s
v a l u e s w i l l be assigned t o
(and t h e r e f o r e t o each of t h e a t ) , cf.9.5.
i n t h e formula (i.e.
at l
a"
If we d e l g t e now
i n t h e graph), f o r 0- a l l members (i.e.
a l l v e r t i c e s , edges and l a b e l s ) except t h e above s t r i n g
ho,hl
,...,hn-,, , t h e n we g e t t h e r e l a t i o n
g* with
g = h h ***
-nu = 3 , i.0 e 1. ,
-*hn-, t@)O
(1 ) 5 M which obviously c o n t a i n s
A
B _ c g e e M * . A s d i s c u s s e d b e f o r e t h e p ~ o o fi s f i n i s h e d .
--
-
Concrete characterization of sub a
$10
10.1
-
It is easy to see that Q p R A )
coincides with [Q]('
)
_-----____-_-
iff Q is an i
_______-__
ntersection structure
(i.e., the intersection of every subfamily of Q belongs to
=[~ER;')
Q)
. Moreover, this implies (by definition) that
Q = LOC [Q] (1
iff Q is an a
--l g--------e b r a i c (cf. e.g.B6n72(3.6.1v)
inter-
section structure (i.e. closed under unions of directed subfamilies of
&, too;
cf. 1.13).
Thus we conclude from 6.2 the well-known result of
G. Birkhoff and 0. ~rink(Di/~r-,
cf.[~6n72(3.6.4)]):
--
(A)10.2
Theorem.
sal algebra iff
.(.-or 9
--
--
L f. 2A is the subalgebra lattice of a univerL
is an algebraic intersection structure
-
equivalently, iff L = LOC [L] (l')).
H
Again by 6.2, we have:
(B)$0.3 Theorem.
7
--
with operations of rank at most
1_11_
structure which
-
-
is
(cf. 1.12), i.e.,
--
2A is the subalgebra lattice of an algebra
L
A
-
s
-
iff
-
closed under unions
iff
is an intersection
-
L
of
a-directed systems
L=~-Loc[LJ(~); sea. H
The equivalent condition 1,.I 4(b),(put Q
=
L) was given by
Go hhrken and rediscovered by lil. Gould[Go68J(cf
.[56n72 (~.94mi
For unary algebras see also fibn72(3.6,7)1, fJoh/s&
The subalgebra systems of algebras of finite type can be
characterized nicely by nearly the same condition (adjoining
a condition on cardinalities only):
The following conditions are
(C)10.4 ~heorem(cf.
)
]
b
2
7
0
~
[
.
.equivalent:
(i)
~(X={A;F)-&Ffinite
(ii)
3 feo,:
L = S U ~ @,
:
~=~nvl')f( = sub<~;f)),
-
--
(iii) There exists an sEIN such that
a) L = s-LOC[L]
(cf. 10.3), or equivalently,
~BCA()~X~B,IXI~S:
rL(~)5I3)+I3~L
-
I I ~ ~ ( x ) I <= No -for all
b)
(For 1 -unary algebras see [Jbn72(3.6.8)].
-
(cf.
1.14)
X c- A with JXILs.
)
The proof follows from a more general result of
P
---_--roof.
M. Gould [~o72b(~.370 u(cf. 12.7):. Neverthelees we give the
prove for this simpler
case:
(ii)j(i)l0*&(iii)
is obvious,
thus we have to prove (iii)+(ii):
For X ={xo,,
ments of
,x,-~] we can enumerate (by '(iii)b)
)
the ele-
rL(x):
(if this set is finite with n elementa take a
:
=
a
;
- for i s j
X
mod n) such that ao=aom
Define the (s+l )dry function
if y=at for x={x0,
f (x0,'",
and let
fl =<~;f).
x ~ 'Y)
- ~:=
Clearly, L g sub
...,xs-11
otherwise,
To prove the inverse,
let
f
~ t ~ u b and
a
(xO'
hence
0"
x=fx0,
X
at = at+l
...,x,-~,$5 B . Because o f
we g e t by i n d u c t i o n on t : I?L (X) 5 B,
,XB-,
L
BoL
by ( i i i ) a ) . Thus
L = S u b a .1
10.5 Remarks.
-------
a ) Since ~ u b < AF>=
;
1nv; ) F , t h e r e n a t u r a l l y
a r i s e s t h e q u e s t i o n how t o c h a r a c t e r i z e t h e s u b a l g e b r a systems
of c a r t e s i a n powers of (A; F
), i. e m, t h e s e t
(
SU~<A;F>' = InvA d p
The answer was g i v e n i n l"fios7,8] by I . G . Hosenberg ( i n t e r m s
of s u b d i r e c t c l o s u r e systems; c f a l s o l s z 7 8 (~hm.g)]).
C l e a r l y , from 6.2 we have a t once :
.
b ) Theorem 10.2 a l s o p r o v i d e s t h e a b s t r a c t c h a r a c t e r i z a t i o n
(i.e. up t o isomorphism^) of s u b a l g e b r a l a t t i c e s ( c f . [ ~ i / ~ r ] ) .
For more s p e c i a l r e s u l t s we r e f e r t o [ ~ o h / ~ e i ] , f ~ 6 n 7 2( s e c t i o n
3.8)J,f~a],[wU*
$1 1
,
Concrete 'characterizationof Con @
Only few results concerning the concrete characterization
problem for Cons can be found in the literature, A partial
solution has been given by M. Arxnbrust
In
R, Quackenbush and B. Wolk proved that any finite distribu(containing the least and the greative sublattice of
test element) is a congruence lattice.
This result can be extended to arbitrary complete distributive sublattices of f(A) as it is done by H. ~ragkoviEova
in ]
r
~
[
and (independently) by S. Burrs, H. Crapo, A.Day, D.
Higgs and W. Nickols (during 1970)(cf. [J6n72(pe1
.
])
4
7
[&I.
AU/WO]
P(A)
-
In [~6n72(~hm.4.4.1 )] 13. J6nsson gave a solution of the
blck
characterization A for ConCI. 11. Werner gave in Die741
the substantially same (but nevertheless better) result by
using so-called graphical compositions.
We can interpret these results as modified versions of the
following theorem, namely as the description of the closure
operator LOC l ~
especially
]
for sets Q of equivalence relations.
--
(~)11: .1 Theorem.
--
C I; f(~) ia the congruence lattice of a uni-
-
versa1 algebra iff C = f(A) n LOC[C]
~(A):~~-Loc[c], cf. 11.2).
.
I
-
(or equivalently C =
I (by 6.2).
(g)11.2 Remark.
_____-It.is well-known
that CO~<A;F)= Con(A;F')
where F 1 is the set of all unary polynomial functions of
{A;F) (lee. F 1=(~v-(oalat~3>(1) ). Therefore the algebra in 1 1 .I
can be chosen always as a unary algebra and, by 6.2, we can
replace LOC [c] by 1 -LOC [c].
,
(21 i s r e f l e x i v e t h e n t h e e q u i v a l e n c e r e l a t i o n
-9If g e 8eLOC[C]
n e r a t e d by €3 b e l o n g s t o ( 1 - ) L O C [ C ] ( C ~ .1 . 1 3 ) , s i n c e B
c a n be e x p r e s s e d as a u n i o n of c o m p o s i t i o n s o f 9 and 9-I.
T h i s g i v e s an l t a l g o r i t h m l l t o prove C = Y(A)n LOC LC] :
Take ~ ~ L o c [ c ] ( ~( )t h e n 0 i s r e f l e x i v e ) and p r o v e B E C
T h i s m u s t be v a l i d f o r a l l s u c h 8.
If C h a s t h e p r o p e r t y t h a t t h e e q u i v a l e n c e r e l a t i o n g e n e r a t e d by u$3il i c 1 ) a l s o b e l o n g s t o C whenever a l l Qi&C,
t h e n t h e f o l l o w i n g c o n d i t i o n d o e s t h e j o b as w e l l :
V~.~G[C]'*) : ~ E C .
.
-
(C)11 . 3 P r o p o s i t i o n .
C 5 &A)
algebra
~ = < A ; F >of f i n i t e
Q = LOC[C]
s a t i s f i e s 6.7(r)
------
-i s t h e congruence l a t t i c e of an
(i.e. P i s f i n i t e ) iff
and
C = P(A) n Q
. I (by 6.7)
I n c o n n e c t i o n w i t h congruence r e l a t i o n s o f u n i v e r Remark.
s a l a l g e b r a s t h e r e a r i s e s t h e problem how t o c h a r a c t e r i z e
( c o n c r e t e ! ) t h e l a t t i c e o f a l l congruence c l a s s e s o f
an a l g e b r a . Be r e f e r t o [ w i l l f o r s u c h a c o n c r e t e c h a r a c t e rization.
a
$1 2 Concrete characterization of ~ u t and sub0
From 6.2 we get:
IA).~2.1
For L G- 2A and G 5 S A , there exists
Theorem.
bra &=(A;F>
_
I
_
L=Q(')
such that
and
L = ~ u b dand G=Aut&
G' = Q n S i where
Q=LOC[LVG'].
an alge-
iff
(6.2)
Because clones of relations are rather complicated we seak
for better conditions combining the results of $8 and $10.
For
(A),12.2
- - 'Rheoremc[~t~U),.
algebra
a =(A;F>
such that
L 5 2A
&
L = sub61
-iff
Gg SA, there is an
and
G = Aut d
are satisfied:
the following.conditions -
--
(i). L is an algebraic intersection structure (or equivalently, L = LOC[L]('
1, cf. 10.2);
--
(ii) G is a locally closed permutation group (i.e.
G=S~"LOC<G> ,(8.1))
A
-- g 4 G and BcL;
(iii) g(B) :=Cg(b)(bt~} belongs & L for all
--
(iv) CG(,B),€L
for all finite B c A
12.3 Remarks.
-------
(notation cf. 8.3).
We mention here some equivalent conditions.
-
-
is equivalent to each
Assume 12.2(i) and (ii):, then 12.2(,iii) -
of the following conditions:
--
( i i i )il
r L ( ~ ( B Z~) g ( , lLl(B))
( i i i ),
g(l?L(~))
L
-
for ~
c
f o r gtG
g ( T ~ ( B ) ) LE
( i i i)
-G-and a l l f i n i t e
( g ( ~ ) ) f o r gtG and a l l f i n i t e B L A;
l?L(g(~)=
) g ( rL(13).)
(iii)3
grG and a l l f i n i t e B s A ;
BsA;
and a l l f i n i t e B S A .
-
Moreover, 1 2 . 2 ( i v ) i s equivalent t o
--
L
I? ( B ) g CG(B) f o r a l l f i n i t e B S A ( i . e . ,
( i v I,,
m u t a t i o n ~o f
i f two p e r -
G a g r e e on B t h e n t h e y a g r e e on r L ( B ) ,
as w e l l ) .
Theorem 12.2 was g i v e n by M.G.
S t o n e i n [st72(~hm.4,
p.46)]
w i t h c o n d i t i o n ( i ~ i)n s~t e a d o f ( i v ) and i n d e p e n d e n t l y by t h e
present a u t h o r (unpublished, with c o n d i t i o n ( i i i ) p i n s t e a d
o f ( . i i i ))
.
-------
of 12.3:
P r o o f
(iii),4+iii),1
(iii)J(iii)4
L
: g ~ g g ( r (B))
obvious.
(iii)4
rL(gB) 5 1 t L ( g ( r L ( ~ ) ) =
g(rL(~)).
i
-1
g
1i
L L
(r r
2B g
-1
L
(I' ( g ~ ) )3 T ~ ( E ) ~( r~L (~g ~( ) c_
)~ - ~
( g ~ ~ (bi i yi l l
ciii)2+(iii):
Let
we have
g( rL(B)) = I ~ ~ ( ~ B ) .
BeL. We show T ~ ( Y ) , ~ ~ f( oBr )a l l f i n i t e
Y s g ( B )(since t h i s implies
5 B
3
g-l ~l?'(Y)~)(~g)~
-
g(B)&,
c f .1 .8,1 . 1 4 ) .
From g-ly
l?L(g-l ( ~ ) )-c l ? ~ ( l =
3 B,
) thus
(iii).
( i i i ) l & ( i i i ) 2 obviously.
(iv)
L
( i v ) j ( i v ) , : B ~ c ~ ( B ) r ( B ) ~ ~ ~ ( c ~ ( . =D )CG(B).
)
i v 1,
(iv),j(iv):
For f i n i t e D g
- CG(B) we g e t D g I? L ( D ) ( 5
-
3
+
cG(D),$ CGCG (,B) = CG (B)(CG i s
I
a c l o s u r e operator! ) , consequently
CG(B),= U { T ~ ( D ) D c C G ( 3 ) & D f i n i t e )
belongs t o
L by ( i )
( c f . 1.13). 1
---------------of 1 2 , 2 :
The ( " i f u - p a r t o f t h e ) proof c a n be done by
Proof
a s t r a i g h t f o r w a r d c o n s t r u c t i o n o f t h e a l g e b r a 0 ( c f .[St72]).
YJe choose a n o t h e r p r o o f ( u s i n g 12.1 ) i n o r d e r t o show a g a i n
how t o work w i t h c l o n e s o f r e l a t i o n s and t h e i r p r o p e r t i e s
-
(because i n o t h e r c a s e s
e.g.
-
7.2
o n l y t h i s neth hod w o r k s ) .
------
P a r t I. The c o n d i t i o n s ( i ) - ( i v ) a r e n e c e s s a r y :
G = A u t B ( , @=(A;F},
If
L=~ubc-,
t h e n ( i ) , ( i i ) , ( i i a r e obvious ( c f . 10.2,
(cf .2.4)
(B)(e*71 CG (B),
5
%OIG
8 . 1 ) , moreover
t h u s ( i v J l (and t h e r e f o r e ( i v ) , c f . 1 2 . 3 ) h o l d s .
P
------a r t 11. The c o n d i t i o n s ( i ) - ( i v ) a r e s u f f i c i e n t .
G o = S i 6LOC [G'U L] ( s i r n i l i a r t o t h e p r o o f
S422-1: We p r o v e
o f 8.1).
Bg A
Let
geSA and
g0hLOC[G'U
t h e r e i s a binary r e l a t i o n
(gIB)'6
sB 5 g ' . Then
yB
L].
Then f o r a l l f i n i t e
gBc&[GouL] s u c h t h a t
'nust be d e f i n e d by a f o r m u l a o f
t h e following type ( c f . 3.2(R4)):
a . e B . ( j e J ) & g k ( a k ( 0 ) ) = a k ( l ) (keK)
( a i i I :
where
J 5 1u{0,lf
J
J
, B.cL,
J
k ( O ) , k ( l ) L I V { O , ~ ~ , glc6G ( J , K , I . i n d e x
s e t s ) . By t h e same a r g u m e n t s a s i n t h e p r o o f o f 8.1
a.
and al must be l l c o n n e c t e d l l , i . e . ,
...,ai n=al
such t h a t
($(0),%(1 ) ). f o r some
vided
BIZ^)
diction t o
gg
there are
( i t ,it+l
) o r (it+l , i t )
kpK (0 5 t
4 n-1
a0=ai0,ail 9
a r e equal t o
) (because o t h e m i e e (pro-
c o u l d n o t be a p a r t i a l 1 - I - f u n c t i o n
ggsg'),.
f = g i g;
o
1
Hence ( g ( B ] * ~ ( f l B ) *where
... glcn-l
&<G>~
A
(p. 53 ) ,
-
7ii)
G
with
i n contra-
Thus
gIB = flB and we g e t
A
~ t e ~ - 2W:e
---
prove
(iiL
gr~oc(~)S
G.
(Q.E.~)
L=Loc[G*uL](').
Let P ~ L O C [ G * V L J ( ' ) and
B be a f i n i t e s u b s e t of g . We
show t h a t t h e r e w i l l be a
B D t [ ~ ] ( l ) = L such t h a t
BSOB53
because t h i s i m p l i e s 9 t LOC [L] ( I ) and we a r e done by ( i ) .
By o u r assumption, t h e r e i s a
gg€[GeuL] ('I such t h a t
B 5 yBC 9, t h u s t h e r e i s a d e f i n i n g formula f o r
SB of t h e
f o l l o w i n g form:
3(ai)iEI:
where
aJ. a J. (jeJ) &
gk(alCcO),),
= ak(l ) ( k m ,
J~IU{Ok
~ (, O ) , k ( l ) e 1 ~ ~ 0 ) .
Consider t h e l a b d e d graph w i t h t h e v e r t i c e s
labled with
with
B
3
(a
j
(ak(0),ak(l) ) labled
f o r j r J ) and edges
ai(i61)
gk f o r ktK. C l e a r l y one can assume t h a t a l l
ai(i&1)
a r e connected w i t h a. (unconnected components do n o t change
(provided t h e y a r e c o n s i s t e n t ) ) . By c o n d i t i o n (iii)one
c a n move a l l v e r t e x - l a b e l s B
j
-
to
a.
(g(a0)=a.eB.
J
g(ao)=
J
& ageg-' ( B j ) ) , and
s i n c e G i s a group and L
j
t h e above formula f o r YB
sed under i n t e r s e c t i o n s
a
-
i s clo-
can be
t r a n s f o r m e d t o t h e f o l l o w i n g form:
where
DtL, ftaG ( t € T i n d e x s e t ) . Define
aB:=
D n cG(B).
Then
B 5 CKg
since
ft(ao)=ft,(ao)
if
Bf
Pg 5 D.
aoCB
and
Moreover, s i n c e
B 5
sBwe
have
i t = i t ,( t , t t ~ T ) .T h i s p r o p e r t y
,
holds not only f o r
CG(B). T h e r f o r e
ft,(a;)
aOtB but
GB5
sB,s i n c e
8.3
- also
a. s
for all
0 -aheyB. Thus B $ Q B $ P B g ~ . l o r e -
i t = i t ,i ,. e .
if
- by
ai)rD n C G ( B ) i m p l i e s
f ( a i)
t
.
o v e r , WB€L by (i) and ( i v ) . Q.E.D.
F i n a l l y , s t e p 1 and s t e p 2 t o g e t h e r f i n i s h t h e proof because
of 12.1.
Analogeously t o 12.1 and 12.2 we g e t c h a r a c t e r i z a t i o n
(2):
theorems f o r t h e I1bounded c a s e "
There e x i s t s a n a l g e b r a w i t h o p e r a t i o n s of
( S ) l 2 . + Theorem.
r a n k a t most
_
I
se7N
-
L=Q )
and
-
0' = Si n Q f o r
For
--
(,B)12.5 Theorem.
algebra
C
such t h a t
--
L=SubCI
and
-
Q = s-LOC[L
U G
w i t h ~ -5 0 i ~ ) ( f goi rv e n S
L=~ubfi and G = A u t &
i f and o n l y
conditions
are
* ~ .
if,
iff
S (6.26b))
-
L g 2A , G 5 SA, t h e r e e x i s t s a
&=(A;&
3
G = A U ~ &
universal
--
L ~ such
)
that
-
respectively) the following
fulfilled:
(i)
L=S-MC[L]
( o f . 10.3);
(ii)
G = SAfi s - L ~ c ( G ) ~
A
G = SAn @+I)-LOC{G)~
(22
the condition f o r G given
-
i n 8.4(in case
----
s 2 2 ) and 8.5(&
case s = 1 ) , respectively);
( i i i ) g ( ~ L ( ~s)y)L ( g ( B ) )
(iv)
--
CG(~),€Lf o r a l l
for
BEA
gbG
-and a l l B F;A
w i t h IaGs.
w i t h (Bige;
12.6Remark.
------
Assume 1 2 . 5 ( i ) and ( i i ) .T h e n c o n d i t i o n
1 2 . 5 ( i i i ) i s equivalent t o 1 2 . 2 ( i i i ) ( c f .
l 2 . 5 ( i ) , and 1.14).
proof o f 12.3, u s e
Moreover, c o n d i t i o n 1 2 . 5 ( i v ) i s e q u i v a l e n t
t o I 2.2(.iv) ( s i n c e C ~ ( B =) N
u{c~(B~B
) /' 5 c ~ ( B &) I B ~ I ~ B J Cby
L
Pro-o f
.
o f 12.5;
(ill,
The c o n d i t i o n s a r e n e c e s s a r y ( t h i s f o l l o w s
from 10.2 f o r ( i ) ; 8.4, 8.5, 8.8 f o r ( i i ) , and from 12.2, 12.6
f o r ( i i i ) ,( i v ) ).
The c o n d i t i o n s a r e s u f f i c i e n t , t o o . I n f a c t ,
S t e--~1 : G o =
---
Sd n s-LOC [G* v L] c a n be proved a s i n s t e p 1 o f t h e
proof o f 12.2 r e p l a c i n g v v f i n i t eBtl by
IIBgA w i t h at most s
element s I t .
S t e----~2 : L = s-LOC
[GO
v L]('
) can be proved a n a l o g e o u s l y ( c f .
proof o f 12.2) o r s h o r t e r a s f o l l o w s : ( i ) - ( i v ) imply 1 2 . 2 ( i ) -
s-LOC[G*
v I,](\'
) = s-LOC L
12.1) + L = [ G ' U ~ ( 1)-4
L](')(by
L=LOC[G'U
( i v ) by 12.6, t h u s
(i2 L,
and 12.4 f i n i s h e s t h e proof .l
-
F i n a l l y we c o n s i d e r t h e " f i n i t e c a s e v 1 (C). The f u l l answer
was g i v e n by M.
--
(,C)I 2.7 Theorem.
Gould i n f i o 7 2 b ( ~ . 370)]:
For an i n t e r s e c t i o n
p e r m u t a t i o n group
structure
L b2A
-and-
-
G=
4SA t h e following conditions a r e equi-
valent :
(I)
L=Sub@
&
G=Auta
f o r -some a l g e b r a
-
G =~ u t
f o r some
--
w i t h f i n i t e F.
(11) L = Sub 61 and
w i t h one
--
a
operation
W=(A;F>
a l g e b r a @(=(A; f>
(111)
- e,neJN
There e x i s t
such t h a t
7-
ForBsA,
(,111.i>
a l l XgB w i t h 1x15s
-
(111.ii)
G
is
n-locally closed, i.e.
( I I 1 . i i i ) g ( rL(13) ) E L
all B cA
-
gcG
and
F ~ ( B CG(~),of.12.3)
) ~
B c-_ A
.
-
with IB)&s
7
(I)+(Ia
by 1 2 . 5 ( t a k e n = s + l ) .
f o l l o w s from M.Gould s proof g i v e n i n [ G O T ~ (~p .
(111) + ( I I )
370)]
for
w i t h (Blce
- ;
1r i v a l .
1
G = SAnn-Loc G ;
I P ~ ( D ) IN,
J
(or e q u i v a l e n t l y
for all
--
------0 f.
and
C~(E)CL
(1II.i.v)
Pro
- for
(i . e . L=s-LOCL, cf.1.14);
B b e l o n g s t o L if ItL (X),cB
by remarking t h a t I!vl.Gouldfs c o n d i t i o n s (1V.i)
,( 1 V . i i )
and ( 1 V . i i i ) f o l l o w from ( I I I . i i ) , ( I I I . i ) and ( I I I . i i i & i v )
above, r e s p . N e v e r t h e l e s e we e k e t c h t h e proof of (III)=+(II):
Proceed a e i n t h e pro05 o f 10.4 but. choose t h e enumeration 09
X
r L ( x ) compatible w i t h G i n t h e s e n s e t h a t g(ai)=afx(cf
.p.67)
f o r a l l grG. T h i s i s p o s s i b l e because of
, hence
l h e n a l l g6G commute w i t h t h e f
L = ~ub(A;f), G
one can choose (as proved i n [~o72a])
that
-
I I
(III.ii)(cf.12.ii)).
d e f i n e d on p. 67
FL (gX)=g( I? L (XI) by
Aut<A; f>. F u r t h e r ,
an o p e r a t i o n f l & O A euch
~ * = R ; ' ) = S U ~ < A ; ~and
~ > G = A U ~ { A ; ~ ~ ) . Then & = { ~ ; ( f , f l J >
f u l f i l l e s c o n d i t i o n ( I ) . One can asswne f and f 1 t o be of t h e
same a r i t y m z l
h(x0,
.Take
***
&'=4;h> with
*-I
,%) * -
f(xo,...,~-,
if
f l ( ~ o , . . . , ~ - l if
)
Then
a1 i s
%=Xo
%fxo
.
t h e a l g e b r a r e q u i r e d i n (11) ( c f . ( D o ~ 2 b ( ~372)]1.
.
m
78
( 1 3.1 )
$1 3
Concrete c h a r a c t e r i z a t i o n of Aut
1)
For
@)1 3.1 Theorem.
a l a e b r a &={A;F>
with F$OA
such t h a t G = A u t e (
--
ancl
-
G*=SffnQ
where
C 5 t(A),
G tSA
Q=LOC[G*UC]
or
fl
and Con
-
t h e r e e x i s t s an
Fg0ls), resp.,(aeI),
C=Confl
iff
-
c = ~ ( A ) ~ Q
or
reap.
Q=S-LOC[G*VC],
No o t h e r r e s u l t s concerning t h i s c a s e
m(6.2)
a r e known t o t h e
a u t h o r except t h e f o l l o w i n g c o n j e c t u r e o f H, Werner g i v e n
i n [we7U(here
.
t h e c h a r a c t e r i z a t i o n of Aut& & Con0 i s s t a -
t e d a s problem 4);:
------------ ------
13.2 Werner ls ~ o n ; ~ e c t u r e ( f i ~ e 7 44-52)]).
(~.
p l e t e s u b l a t t i c e of
G
Let
C
be a com-
~ C A
( =) e q u i v a l e n c e r e l a t i o n s on A) and
a p e r m u t a t i o n group on A. There i s an a l g e b r a ~ = ( A ; F >
C = Con@
such t h a t
and
G = ~ u t a iff
i s c l o a e d under P +
(we w i l l n o t f o r m u l a t e t h i s
A ?X,Y
c o n d i t i o n e x p l i c i t e l y b u t we n o t e , t h a t i t i s e q u i v a l e n t
(a) C
t o t h e e x i s t e n c e o f a n a l g e b r a flf w i t h C = Con
(b). G
i s l o c a l l y c l o s e d ( c f . 8.1
e x i s t e n c e of a n a l g e b r a
( c ) If
geG
where
and
Qg =
8EC
with
1
i s equivalent t o the
G = Aut &It)..
eg€C
then
{lg(x) , g ( y ) )
, this
(x,y)c@3.
at ( c f . $11,)).
Clearly the conditions are necessary but it turns out
that they are not sufficient.
In the preceding paragraphs we applied the General Galois
theory (.inparticular 6.2), to special cases in order to obtain ftgoodff
general characterization results. Now we become
acquainted with another application of our theory: the construction of counterexamples.
1.3.3 Counterexample
----------- -- to 13.2.
A=P
Let
= {0,1
,2,3f and
),(I ,0),(1 ,I >,(2,2),(2,3),(3,2),(3,3)J
Q ={(0,0),(0,1
,{2,3j),
partition {{0,1)
),(I
Q'={0~0),(0,2),(2,0),(2,2),(1
9 partition {{0,2f
Qo={(x,x)
l xtAJ.
el= A x A
9
,3),(3,1),(3,3))
,$I ,351,
9
g = (03)(12)€sA, e =identity of SA,
c ={Q~,~,Q~,~~),
G = l e ,g3.
Then 13.2(a).,(b),(c)
C=CO~(A;
are fulfilled for C
and
g,h> with h(O)=h(l 112, h(2)=h(3)=0;
G
(e.g.,
and note that
- S A is locally closed), but there is no
for finite A, every G 4
algebra (X with
C = cone(
and
G.= Aut fl
.
To see this, consider the relation g defined by
(ao,al1 4 3 :-
3 a GA:
lao,a ),re & (a0,a)eQ
It is easy to check that
f=
fore fm€S~n[~*uc](cf.3.5).
then fiG by 13.1
f' where
'
g(ag) = a
f = (01) (23)&SA. There-
If the algebra & would exist
, contradiction!
13.4
-
P o s s i b l y W e r n e r ' s c o n j e c t u r e becomes t r u e i f c o n d i t i o n
( b ) w i l l b e r e p l a c e d by a s t r o n g e r one; e.g.
we formulate t h e
f o l l o w i n g c--o n J -----ecture:
; ~= (A; F>
2nd G L SA* There 2s _an ~ & @ ; g b8(
= Cons and
iff
--- G = Aut@ ---
he4
C $ C(A);
s u c h t---hat C
---(a)
3 flf=<A;l?l):
(bi)
G* = S
;
'I L O C [ G ' ~
VWC
(c)
C =Con @I ( c f . 1 3 . 2 ( a ) ,
:
~f
eg€c
I 1 .I );
( t h i s implies 13.2(b));
(cf. 13.2(c)).
Comparing t h i s c o n j e c t u r e w i t h 13.1
, the
a d v a n t a g e of 13.4
c o n s i s t s i n t h e f o l l o w i n g : The o n l y i n f l u e n c e o f G o n C i s
g i v e n by c o n d i t i o n ( c ) .
C l e a r l y , a f t e r p r o v i n g ( i f p o s s i b l e ) t h i s c o n j e c t u r e one
s h o u l d l o o k f o r .a s i m p l e r c o n d i t i o n ( b ' )
If
i s t h e t r i v i a l congruence l a t t i c e , i . e .
C
algebra
.
&
i s r e q u i r e d t o be o i m p l e , then13.1
if the
provides a
f u l l answers
-
13.5 ~ r o ~ o s i t i o n ( ,.c[ fS C ~ I ~ . E . T . ~ ~ ) L
. et
-a s i m p l e
fl =(A;F) w i t h
G = L O C < G ) ~ n sA .
algebra
G g S A . There e x i s t s
--
-
G = A U ~ Q i f and o n l y i f
A
-------
Proof.
We have
LOC[G'VC]=
LOC~G']
if
C
Q I
={$2,6z5,
i.e.,
if
C
1
c o n s i s t s o f t r i v i a l congruence r e l a t i o n s o n l y ( n o t e , ( X , ~ ) G ~
n
c a n be r e p l a c e d by
x=y
, and
(X,~)C~:
c a n be d e l e t e d i n
e a c h f o r m u l a which d e f i n e s a r e l a t i o n o f [ G ' V c]).
Thus, by 13.1 and 0 . 2 ,
8 . 1 , we a r e done i f
C =
C(A)~ L O C ~ G * ] .
T h i s c a n be shown w i t h o u t d i f f i c u l t i e s ( u s i n g e . g o 1.13 and
o r proceed a n a 1 o ~ ; ~ o u o las
y i n proof of 8 . 1 ) .
3.5
But t h e r e i s a l s o a sirnple d i r e c t p r o o f . C l e a r l y , t h e condition
G =
s A n LOC(G)
Take
i s n e c e s s a r y ( c f . 8.1 ). ETow l e t
A
By 8.1 t h e r e i s a n ( ~ ( ' = ( ~ ; ~ p w iAut
t h &'=G.
G = SAn LOC(G)~
8( =(A;
.
S~
~ u i t j >where
t(x,y,z) =
Because
Aut (A;
x
z
t
i f xfy
i f x=y
't)= SA and (A;
flfe78(Lemma 1 . 1 0 u ) ,
&
i s the ternary discriminator
.
t)
i s s i m p l e ( c f . e.g.
i s a l s o s i m p l e and
A u t o = 0.
I
82
(14.1
$1 4
C o n c r e t e c h a r a c t e r i z a t i o n of End(% and sub&
Thia c o n c r e t e c h a r a c t e r i z a t i o n problem was ~ l o l v e di n
[~a/~t77a] i n teriils o f systei:ls of e q u a t i o n s wi.vilicil r e f i e c t
- in
-
our terminology
t h e p r o p e r t i e s of
L O C [ ~ I * V L].
From 6.2 we g e t t h e f o l l o w i n g theorem:
--
algebra
& =(A;$
such t h a t
--
-
Par
(B)14.1 Theorem.
('I and
H50A
Lg2A, there exists 9
with
(gl F C O A Or
(21 F 5 0 A('1 ( S E X T I , r e s p . ,
H = E ~ ~and
B L = S U ~ L Y iff (foT
-
Q=H0uL )
H = ( O ~ ) ) * ~ L O C LL =QL~O, C ~ Q ~ (o' r)
(A)
-
( B ) . H = ( O ~ ) ) ' ~ ~ - L O CL ~= Q
s - ~L ,~ ~ [ ~ ] ( ~ ) , r e sDp .
-
The I f f i n i t e c a s e t l ( C ) ( c f . 7.1 ) might be t r e a t e d w i t h 6.7
-
( o r w i t h 14.1.(B) f o r f i n i t e A), b u t no c r i t e r i o n ( f o r
( 2 ) ) is
known t o t h e a u t h o r which u s e s p r o p e r t i e s o f H and L o n l y .
How theorem 1 4.1 might be improved (e .g.
analog3ously t o
12.2 i n cornparision w i t h 12.1)? We mention h e r e some n e c e s =
sary condition:
14.2 P r o p o s i t i o n .
univeraal algebra
are satisfied:
-
If
H=Endfl
8 =<A;F>
and
-
-then t h e
L=sub&
f o r some
-7
following conditions
L
is an a l g e b r a i c
i n t e r s e c t i o n s t r u c t u r e ( c f . 10.2);
H * = ( o ~ ) ) * ~ L o c ~ (~c fI. * 9.1);
]
and BeL (i n particular
t h e image h(A) of h belongs to L -for a l l ~ c H ) ;
he' ( ~ ) = { a sh~(/ a ) ~E ~
L j-f o r a l l hcII and BeL ;
--
h(B)={h(b)l ~ C B ! E L f o r a l l
<a)€ L
ca€H
PPol &B)EL
(where
hcB
ca :A--+A:
-for a l l finite
for a l l finite
--
BsA
r L ( B ) = C ~ ( B Jf o r
x C.+ a ) ;
BSA ;
;
all
B ~ A .
P
---.--I
roof.
( a ) , ( b ) a r e obvious(cf. 10.2, 9.1).
( c ) and ( d ) can
be shown by a s t r a i g h t f o r e w a r d proof ( c f . 1 2 . 2 ( i i i ) ) .
obvious.
( f ) and ( g ) a r e analogeously t o 1 2 . 2 ( i v ) ( c f . 8 . 7 ) :
We have TPol $B)E[II'](~.)E
- LOC[H*U
F u r t h e r , f o r al
11'' ) = L
,..., a n ) ) = h l ( f ( a l ,..., a n ) ) , i . e .
-
by 4.33 and 14.1A.
,...,an&CH(B), f & F ( " ) , we g e t a = f ( a l ,...,a n
CH(B) s i n c e hIg=h1IB = $ f ( h a l ,...,han)=f(hlal
h(f(a,
(e) is
,...,hla,)
)E
+
h ( a ) = b l( a ) .
Condition ( h ) i s e q u i v a l e n t t o ( f ) ( f o r t h e proof s e e 1 2 m 3 ( i ~ ) , ) m
I
Remark.
--.)----
It i s unknown t o t h e a u t h o r whether c o n d i t i o n s
1 4 . 2 ( a ) - ( h ) a r e s u f f i c i e n t , too. ( p o s s i b l y under c e r t a i n r e strictions to
H
and/or
It i s easy t o s e e t h a t
L ).
TPol$B)gCH(B).
It i s not c l e a r
i n which c a s e s e q u a l i t y holds ( c f . 8.7 f o r groups).
Probably (g) f o l l o w s from ( f ) and (a).
$15
Concrete characterizations IV.
(Survey on related Galois-connections)
With the preceding paragraph we close our considerations
of special concrete characterization problems. Of course,
there axe some more problems than treated in $58-14 (e.g.,
~ n&
d & Con@
, Sub
zation of Auto
& con@ )
, ~ndfl, sub &
. The simultaneous characteriand con@
was given in $7.
The common background of all these results was the description of Galois closed sets of operations or relations
with respect to the Galois connection Pol-Inv. In the most
cases this Galois connection was restricted to special kinds
of operations or relations, resp. Let us sketch once more
this treatment in general:
15.1
E soA and
Let
El
p RA be sets of operations and re-
lations with given "propertiesu, resp..Then the operators
F-Kt(F),:=
Q
W K(Q):=
EtfiInvAF and
E
fi
PolAQ
(FsE, Q$Et)
define a Galois connection between subsets of E and Et.
Suppose we have characterized the Galois closed sets
(O)
F = K(Kt (F)) ( = E nPol(Etn Inv F))
(81
Q=K'(K(Q))
( = E t ~ I n v ( E n P o Q))
l
and
.
Then we have (obviously) the following characterization
(+)
For
-
F cE,
- there exists a relational algebra (A;Q)
--
Q f E' such that F = E n P o l Q
iff
F satisfies
fO).
with
(**)
For
Qc
-Et , there
FSE
-such t h a t
22
-
u n i v e r s a l a l g e b r a (A; F) w i t h
Q = E ' A I ~ V
F
iff
Q
-------
(81,
-
Q i s G a l o i s -.~lo*.
(-i . e
P r o o f.
satisfies
.
( r c ) : C l e a r l y , F=K(Q) =3 K ( K t ( F ) ) = K ( K 1(K(Q),))=K(Q)=F.
C o n v e r s e l y , if F=K(K1(F)) t h e n d e f i n e &=K1(F).
(*+)
can b e proved a n a l o g e o u s l y .
C o n v e r s e l y , e v e r y c h a r a c t e r i z a t i o n theorem o f k i n d (+) o r
(m*) i s (more o r l e s s i m p l i c i t e l y ) a c h a r a c t e r i z a t i o n o f
Galois closed s e t s o f operations o r r e l e t i o n s , r e s p e c t i v e l y ,
where t h e G a l o i s c o n n e c t i o n u n d e r c o n s i d e r a t i o n i s g i v e n by
the operators
that
Pol-Inv
m o d i f i e d a a above. Note f o r example,
I n v P , Sub F ( = ~ n v ( ' ) F ) , Con F, P o l F
End Q ( = P o l (' ) Q ) , w-Aut
Et n Inv F
1 5.2
-
and
Pol Q,
c a n be e x p r e s s e d i n t h e form
Q
E nPol Q
and
respectively.
I n t h e f o l l o w i n g t a b l e we summarize a l m o s t a l l r e s u l t s
given i n previous paragraphs under t h e p o i n t of view o f
G a l o i s c o n n e c t i o n s ( c f . 15.1 1. We r e f e r t o t h e remark a f t e r
to
4.2 ( a n d 4.1b, 4.2, §$8-14) f o r some r e f e r e n c e s c o n c e r n i n g
G a l o i s c l o s e d s e t s o f r e l a t i o n s (sometimes r e s t r i c t e d t o oper a t i o n s ; we add h e r e [ I s k ] ( ~ o l ~ ~ o nf ~
o rF p - r i n g s (A; F> and
[~ie](~ut 1nv'")~ f o r G
6 SA)).
I n t h e t a b l e , r e s u l t s on
w-Aut
F
and
Aut F
considered
i n t h e n e x t p a r a g r a p h a r e m e n t i o n e d , t o o . Note t h a t w-Aut F
f o r F G O ( c f . 1 . 6 d ) , t h e r e f o r e no d i s t i n c t i o n i s
- A
needed i n some c a s e s .
= Aut F
The t a b l e g i v e s t h e number o f t h e theorem i n which t h e
K1(K2(U)) o r
Galois closed s e t s
K1(K2(F)) ( 9 5 R A , F e O A )
were c h a r a c t e r i z e d ( o r from which t h i s c h a r a c t e r i z a t i o n
immediately f o l l o w s ; have 1 5.1 i n mind! )
-7-K2=
El
.
--
-
+
/
,-
Inv
Inv ( s )
P O ~
POI(')
( & End
-.
/
,
K1=
,
Inv
,
1nv(']
-
_
-
-
-
-
I
6.2(a)
_
.
_
Con
6.2(a)
1 I .I-
-
Pol
pol(")
End
4 . l ( a ) 4.1 ( b )
4 * 4 - - - - - 5.2$,
A U ~
(4.2)
7.3
4.1 ( a ) 4.1 ( b ) 4.2
5.24)
5.2 p ) . 7 . 3
5.54
5.5 ( 3 )
gal
w- Au-t;
- -..-4.6
5.5 p)
5.545.2 (remark )
-
6.2(a)
8.1
16.6
6.2(b)
10.3
11 .I
11 02
6.2(b)
9.3(ii)
--
1
6.2(b)
9.2
6,2(b)
8.4
8.5
8.6
L--..-
Table: The G a l o i s c l o s u r e
8.8
K1 (K2 ( Q ) ) ( f o r
w a s c h a r a c t e r i z e d i n El
*
pol(l-l
n-~ut
~ u t
- - .10.5.-
Sub
-
.
4 * 2 ( a ) 4.2(b)
4.5
. 6.2(b)
6 . 2 ( a ) 16.2
-- --
.
Q g RA o r O A )
1 6
116.1
-
Krasner-clones of r e l a t i o n s
The s e t s
Q = Inv H
of i n v a r i a n t r e l a t i o n s of a s e t
) o f unary o p e r a t i o n s a r e c h a r a c t e r i z e d by
H 0;
( c f . 4.2(b) ). The G a l o i s c l o s e d s e t s Q = 1 -LOC[Q]=
(with r e s p e c t t o t h e G a l o i s connection
1- L O C [ ~ = g
Inv End Q
I n v - E n d ) sometimes
a r e c a l l e d Krasner-algebras of 1 st kind ( [ ~ o / ~ a l . [. ,~ 6 / ~ a u ) .
Here we w i l l use t h e name
k-----ind
K
r a s n e r - c 1o n e
--------------------
---
o
f 1 st
---
f o r c l o n e s of r e l a t i o n s (3.5) which a d d i t i o n a l l y a r e
c l o s e d under a r b i t r a r y unions ( c f . 1 .I 3 ) .
16.2 ~ro~osition(cf.[~~/~al(1.3.1,
t h e following
-
1.3.411).
-
For
-
QcRA,
conditions are equivalent:
--
(i)
Q i a a Krasner-clone
- -of
- -l's t k i n d
(ii)
Q = I n v End Q ;
3
(iii)
~$0:'):
Q = I n v 13
.
I(4.2(b) f o r s = l ).
Now, l e t u s c o n s i d e r t h e G a l o i s connection
w-AuC), i n p a r t i c u l a r t h e G a l o i s c l o s e d s e t s
I n v w-hut Q ) f o r
QgRA ( f o r
Aut I n v G
I n v - Aut ( o r
I n v Aut Q ( o r
s e e 4.6).
Clearly
t h e s e s e t s must be Icrasner-clones of 1 st k i n d , but t h e y have
t o s a t i s f y some more c o n d i t i o n s , too. We g e t as a f i r a t observation:
1 6.3 Lemma.
-
-
( i ) For
f
feSA
preserves p
P ~ ~ p -i mhave:
)we
++
f-I
zeserves
~q
( := A~ \
p )
(ii) F o-r a p e r m u t a t i o n g r o u x
under
-
(iii)If
(-i.e.,
7
-------
P r o o f.
under
?
9 -
Aut Q = w-Aut Q
then
( i )f o l l o w s from t h e d e f i n i t i o n s ;
r e c t l y from (,i)
(e.g.
V g € Q : t p rI n v f-'
i s closed
g C Q =p 7 5 r ~ ) .
i s closed
Q 5 RA
Q = InvAG
G $ SA,
(iii):few-Aut
+ V V ~ QF e: I n v
.
(ii).,
(iii)d i -
Q + V g e ~ : ycInv f
f-I + f-5 w - ~ u t Q,
$1)
i.e.
f€Aut 9 ) . 1
For f i n i t e
A , t h e p r o p e r t y t o be ( a c l o n e of r e l a t i o n s
and) c l o s e d under
1
i s s t r o n g enough t o c h a r a c t e r i z e t h e i n -
v a r i a n t r e l a t i o n s o f permutation groups. One c o u l d expect
t h a t t h i s i s t r u e i n g e n e r a l . Before d i s c u ~ s i n gt h i s conject u r e we i n t r o d u c e t h e f o l l o w i n g n o t i o n s :
16.4 D e f i n i t i o n .
c------l o n e
o f
--I
Q 5 RA
A set
2nd k-----i n d
---
I<
ra sner ------------
is called a
if
a ) Q i s a Krasner-clone of I st k i n d (i.e. Q=1-LOC [Q] ) and
b.)
Q
i s closed w i t h respect t o
a t r o n~ s
u 2 --. e rp----------o s i t i o n,
---------
i , e . ( f o r n o t a t i o n c f . 3 . 2 ( ~ 4 ) , 3.4):
-
Ti: mi+
A, ieI(inde:r
s e t ) and
1
For
x:m -+
( p i ) i c 1 : = { ~ a aenA &
a l s o belongs t o Q
c ) Q i s c l o s e d under
3t
,
7 ,
1
Yjt-Q ( m i ).9
A, t h e r e l a t i o n
a is bijectiv &
Xis e pi
and
i.e.
yeQ
+7
9 Q~
) Note, arAA c a n be c o n s i d e r e d as a mapping
*,
.
a: A +A
:
i ~ a ( i )
For Q s R A ,
1.6.5 P r o p o s i t i o n .
consider
the f o l l o w i n g
conditions:
of l st -'
kind-
--
o f 2nd ',,k i n d *
(KC1 2 )
Q i s a Kmsner-clone
(KC1 1 ; )
Q i s a Krasner-clone
(C1)
& is a c l o n e of -----.-'
relations*
-
-,-
- -- Q i s closed under i ;
Q c o n t a i n s -t.h e i n e q u a l i t y -r- e l a t i o n 9 ={(x,y) I xfy};
Q i s c l o s e d -.-.--w i t h w e -c-..-t -.-.
t o -s t r o n g -s u p e r p - o s i t i o n .
( 7 )
(9)
( SSUP)
Then t h e f o l l o w i n g . e q u i----v a l e n c e s --and i m p l i c a t i o n s hold:
---
P
------r o o f.
( + ) by d e f i n i t i o n .
,(n
(79.)).
iel
1
-------
Remarks.
(+HI i s c l e a r s i n c e
The c o n v e r s e o f
Example: H = P o l g('-l)3Nt
H3f :
x C , x+l
(-ti-)h o l d s
(+tt)
because of
11
1
u 9.
1
iciI
=
i,=7d<( d ) F = { ( x , x ) l x c ~ )
i s not t r u e i n general.
p r e s e r v e s v and
does not preserve
7
Wt=lN\.[lj,
mt=jiI
. Take
but
Q = InvgH.
I
F o r f i n i t e A ( ( A J 9 3 ) we have ( c f . [ ~ ~ / 1 < a 1 (.3.5)])
1
(KC1 2 ) -
Krasner-clones
(C1) PC
( 7 )
(C1)
( 3 ) .
Q of 2nd k i n d s a t i s f y ( C l ) ,( s S u p ) ,( 7
) and
( v ) . The n e x t theorern c l a r i f i e s which c o n d i t i o n s c h a r a c t e r i z e G a l o i s c l o s e d s e t s o f which G a l o i s c o n n e c t i o n :
---1 6 . 6
or
-
Theorem.
For
-
(a), r e s p e c t i v e l y ,
Q = R A , t h e c o n d i t i o n s ,&ven
---i n ( g ) ,
a r e equivalent:
-
1) & (
(i) 1
,
9
l S t k i n d and
---
i e
2
is a Krasner-clone
of
--.
-JEQ;
( i i ) Q = I n v P o l ('-'IQ
(iii) There
Q
(notation c f . 1.4);
2 set H o f unary i n j e c t i v e mappinm
such t h a t
--
Q = InvAH
(115
- 0':
-
( i ) '( K C 1 1 ) 8r. Q s ~ u p )i ,. e .
of lSt kind
----
))
.
9 i s-a Krasner-clone.-
and c l o s e d under s t r o n ~superpo-
sition;
( i i ) 'Q = I n v w-hut
Q ;
( i i i ) '3 H -C S A : Q = I n v H
( i ) l l
(KC1 2 )
(or ( K C 1
.
1 ) & ( s ~ u p &) (
of znd
-a Krasner-clone ( i i ) "Q = I n v Aut Q ;
( i i i ) °3 G L SA : 9 = I n v G
-
i.e.
)),
kind (cf
. 16.5)
Q
2
;
.
-------
Proof.
(oC): (ii)+(iii
I trivial.
(iii)+(i)
:
of
(P):
by
9 d Q,
(
feEnd Q
ii
i
(4).Moreover,
1,6.4b)) b e l o n g s t o
is bijective
af
(i)+?(iii):
Q = I n v End Q
every
by 16.2.
must be i n j e c t i v e , i.e.
trivia.
=+Ziafcpi
is bijective (since f
Because
End Q =
(iii)'=+(i)
' : & sat.isfies (KC1 1)
f o r g i t I n v H, a l s o
Inv H
i s a Krasner-
9 i s i n v a r i a n t f o r every
c l o n e of 1 st k i n d by 16.2. TJoreover,
i n j e c t i v e mapping.
Q = Inv II
because
for
q:=/kRni(pi)
'Kacp
+
f f H (since f
b i j e c t i v e ) hence
p r e s e r v e s 9;c o n s e q u e n t l y 9 ~ I n vH.
TjaLyi
(of.
and a
preserves pi);
KafGp
, i.e.
f
(
)
i
?=
bE9
1-LOC Q
f
: Let p a ~ n v ( m ) w - ~ uQ.t We show i € Q .
I? H ( b )
rH(b) Q
o f 4.3:
for
H=w-Aut Q ( c f . 1 . 8 ,
2.4).
Clearly,
Since
Q =
i s c l o s e d u n d e r a r b i t r a r y u n i o n s , we a r e done i f
for all
b b ~ The
~ . proof g o e s a n a l o g a o u s l y t o t h a t
d
1
I f H ( b ) (2*4)={f ( b ) e ~ ~ ( f t r r - ~
Qut
=ff(b)lf bijective & f t ~ o l ( ' ) & )
={bbaet~~la: A-A
={ba
belongs t o
& a
Q by 16.4b)
(r: 3 - - - + A ) e p
(a):
I at^^
and
b i j e c t i v e & a t P o 1 Q)
bijective & k e p r Q :
(note, rarpmeans
(1) and
(i)"=+(ii)"
by
-------
f(r)ey for
a = f c S A , c f . 3.4).
i i f i i i ft r i v i a l .
Remarks.
rasp3
i i i f ( i bfy
16.3(iii).
(p)
and 1 6 . 3 ( i i ) .
I
~ ~-----reo b n
lern:
- ---
We s t a t e h e r e t h e f o l l o v i i n g ~
[(KC~ 2)
+=+ (C1) Sc
( 7 )
7 )
The a u t h o r w a s u n a b l e t o prove t h i s e q u i v a l e n c e ( c f . 1 6 . 5 ) ,
which h o l d s f o r f i n i t e
(Cl)&(?).
A , o r t o g i v e a counterexample t h a t
do n o t imply ( K C 1 2 ) . The c r u c i a l p o i n t c o n s i s t s
i n p r o v i n g whether
holds f o r a l l
Q with ( C l ) & (
7
) ( b e c a u s e t h i s would imply
Loc
& f I n v Aut Q = I n v l Z E ( ~ u t& ) f I n v p o l (1 -1
IQ -
Q).
F o r i n v e s t i g a t i o n s o f Mrasner-clones we a l s o r e f e r t o
fir501 ,[~r66],[Kr68l,[Kr76a],[Le76],f~e77]
[ p o i 7 9 ,[~oi80](for
,[Rr/~oi],[~oi71],
f i n i t e A see a l s o fio/~al], fi~/Kal]).
r/;u~\-&&
'u&.m3dH$o:?)*
N O , to-,.
/flurVE& :
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--
SUBJECT'I N D E X
Adjoining f i c t i v e
c o o r d i n a t e s 26
algebra
-, f u l l f u n c t i o n 20
-, Menger 39
-, r e l a t i o n a l 11
-, u n i v e r s a l 11
a l g e b r a i c c l o s u r e 58
algebraic intersection
s t r u c t u r e 66
s - a l g e b r a i c monoid 60
automorphism 13
-, weak 1 3
B d c e n t r a l i z e r 50
i n t e r s e c t i o n a t r u c t u r e 66
-, a l g e b r a i c 66
i n v a r i a n t 12, 1 3
Krasner-clone of 1 st k i n d 87
- of znd k i n d 88
Local c l o s u r e 1 5 , 1 6
l o c a l l y i n v e r t i b l e monoid 60
Permutation o f c o o r d i n a t e s 25
polymorphism 1 2 , 1 3
p r e s e r v e 12
-, s t r o n g l y 1 4
p r o j e c t i o n 20
- o n t o c o o r d i n a t e s 26
C h a r a c t e r i z a t i o n problem,
c o n c r e t e 37, 43, 48
R e l a t i o n 11
c l o n e o f o p e r a t i o n s 20
-, d i a g o n a l 25
c l o n e o f r e l a t i o n s 28
-, i n v a r i a n t 1 2
composition 20 ( f o r o p e r a t i o n s ) , ,
26.(for r e l a t i o n s )
-, n o n t r i v i a l 25
-, O r i v i a l 25
containement p r o p e r t y 1.8
D e l a i n g of c o o r d i n a t e s 26
d i a g o n a l r e l a t i o n 25
d i r e c t e d s e t 1.7
- -, S- 1 8
d o u b l i n g of c o o r d i n a t e s 26
Endornorphiam 1 3
General G a l o i s t h e o r y 5
I d e n t i f i c a t i o n of
c o o r d i n a t e s 26
S u b s t i t u t i o n 25
s u p e r p o s i t i o n 20 ( f o r oper a t i o n s ) , 27 ( f o r r e l a t i o n s )
-, g e n e r a l 27
-, s p e c i a l 23
-, s t r o n g 88
Werner's c o n j e c t u r e 78
INDEX OF NOTATIONS
s~
Sub
13
13948
nm
27
[Q~~,s[Q]
28
LOC/
36
1
[I/-11
39
39
1nvo0
46
I
Con
48
i
f"
Pol
13
Inv
13
End
13,40948
), 1 3
(~)0(1)
A
d2
m
)
21
25
Aut
The c a r d i n a l i t y of a s e t A i s denoted by l A l ; No i s
t h e l e a s t i n f i n i t e c a r d i n a l number ( ip,=IlNl);
2A s t a n d s f o r
U
t h e s e t ( l a t t i c e ) o f a l l s u b s e t s o f A; f o r f : An-+A
and
B GA, f l ~d e n o t e s t h e r e s t i c t i o n o f f t o B ( flB: B n + A ,
(fl~)=
' f e n (BnxA)); An i s t h e nth c a r t e s i a n power o f A ;
AxB denotes t h e c a r t e s i a n product; t h e l o g i c a l s i g n s
a r e used i n t h e u s u a l s e n s e .
j,v,&
Author's address
D r . R. P8schel
Akademie der.Wissenschaften der DDR
Zentralinstitut fur
Mathematik und Mechanik
DDR
- 108. 39Berlin
Mohrenstr