L C M 6 V 2 0- 1 0 / 3 t / 8 8
96. Category Logic and Horn Sentence Translations.
l{e now define two first-order
logical
languages for categories.
By taking conjunctions of the atomic formulas of these languages,
form category diagrams with variables
we can implicitly
objects and morphisms.
Using the first
and exactness relations
can be imposed in exact additive
and Z-Iinear
categories,
In the second Ianguage, the atomic predicates
express diagram properties
for the structures
of almost strongly
our logical
the set llo. of all
J . a n g u a g e sf o r
almost strongly exact relation
UAc(R) denote the set of just
Horn sentences for
Given a ring R, let
Iet URc(R)denote the subset of U*,
for R-Rel.
EssentialJ.y, these
diagram-chasing properties
R-ReI to be represented by formulas.
are restricted
we determine
those sentences in tlo, which are
of sentences which are satisfied
allow certain
basic universal
categories.
for R-Mod. Similarly,
formalizations
categories,
Horn sentences for exact additive
basic universal
categories and the set Ua, of all
of R-Mod and
Since sentences are finite,
to diagräms with finitely
rA (additive relation
algebras) and r,
were already discussed.
basic universal
(additive relation
(iattices)
algebras with
llo and U, denote the sets of all
Horn sentences for each of these algebraic
the sentences satisfied
quasivarieties.
Let ll'
we
many elementary hypotheses.
In $4, universal Horn sentences for the algebraic types r,
unit)
exact
categories.
By defining
satisfied
language, commutativity
combinations of morphisms in each Homset can be
constructed indirectly.
relation
representing
types.
Here,
for R-modules are determined by the algebraic
That is,
UL(R) denotes the subset of 11,consisting
of exactly those basic universal Horn sentenceswhich are satisfied
in every
m e m b e ro f J ( R ) , a n d s i m i J . a r l y f o r l . l o ( R ) a n d 0 ( R ) , a n d f o r U B ( R ) a n d E ( R ) .
Our objective in this
Horn sentence translation
section is to construct a closed Ioop of recursive
functions
T4
T1
T)
Tn
Tn
ün.iün--j_ür.4ilr
c
iün
c4ta,
which are to be defined independently of the choice of any ring.
ring R, however, the set of basic universal
nontrivial
i = 0,!,2,9,4,
To
substituting
define our logical
precisely
T. (f ) e llr(R) for
the appropriate labels for Y and Z in each case.
construct recursive translation
explicitly
Horn sentences
be preserved. That is, f e tlr(R) iff
satisf ied for R will
For each
we must
functions,
languages for categories.
These languages are
c u m b e r s o m e ,a n d n o t v e r y u s e f u l f o r w o r k i n g w i t h c a t e g o r y d i a g r a m s d i r e c t l y .
By examining the formalizations,
however, we can see which diagram-chasing
problems correspond to basic universal
We begin with the logical
6.1. Definitions.
Horn sentences.
Let ro. denote the structure type having a unary
predieate Objecto'
binary predicates equality (=) and Unito'
ternary predicates Morphismo, and Zeroo,
arity
four,
) . a n g u a g eo f a f i r s t - o r d e r
six.
and a
Let L(ro, ) denote the
with
Note that L(rlc ) is a purely
language, without any constants or functions.
first-order
The only terms of L(ror)
are variables.
set of variables X = {*r,X2,X3,...}
for
five,
predicate calculus with equality
atomic predicates of type ror.
for L(ror)
a predicate Negativeo, of
predicates Sumo, and ExactAc of arity
predicate Compositiono, of arity
relational
categories.
language for exact additive
F i x a d e n u m e r a b l yi n f i n i t e
for L(toc).
are constructed by providing
The only atomic formulas
of variables
an argument list
in X
one of the nine atomic predicates given above.
0, is an exact additive
If
category, then each variable
r e p r e s e n t e i t h e r a n o b j e c t o r a m o r p h i s mo f 0 .
we will
(To avoid complications,
assumethat no element is both an object and a morphism of 0,
This is easily arranged for module categories. )
d:X-Objects(0)
is interpreted
below.
of X may
uMorphisms(0) (disjoint
A function
union)
as true for an atomic formula according to the specifications
Let A, B, C, f,
B, h, x and y denote arbitrary
elements of X.
x = y
d(x) = 41t; (anY variables)
O b j e c t o ,( A )
d(A) is an object of 0
Morphismor(f,A,B)
d(f)
i s a m o r p h i s mf r o m d ( A ) i n t o
d(B) in 0
Unitor(f,A)
d(f) = 1a(ol in 0
C o m p o s i t i o n o( f, , g , h , A ,B , C)
d(f ) = d(e)d(h) for d(e):d(A)---_d(B)
and d(h) : d( B )....-d( C ) in 0
d(f) = d(g) +d(h) for d(e) and d(h)
Sumor(f,g,h,A,B)
from d(A) into d(B) in 0
d(f ) = -d(e) for d(g):d(A)*d(B)
Negativeor(f,g,A,B)
i n 0
Zeroor(f,A,B)
d(f)
i s t h e z e r o m o r p h i s mf r o m d ( A )
into d(B) in 0
(d(f),d(g))
Exactor(f,g,A,B,C)
i s a n e x a c t p a i r o f m o r p h i s m sf o r
d(f):d(A)-618)
and d(e),d(B)-d(c)
A, B or C above is intended to represent an object,
Any argument labelled
and is called an object argument of the atomic predicate
arguments labelled
f,
Also,
in question.
g or h above are intended to represent morphisms, and
are called morphism arguments.
Note that atomic predicates having one or more
morphism arguments also have object
codomain of each morphism argument.
arguments corresponding to the domain and
Specifically,
each morphism argument x.
(x.,x.i,x1),
for an atomic formula has a corresponding morphismtriple
"j
in 0
and xo represent the domain and codomain objects for x..
where
For example,
N e g a t i v e o c ( x g , X ? , * 1 , x n ) h a s t h e o b j e c t a r g u m e n t sx , a n d x n , m o r p h i s m
argument x, with morphism triple
m o r p h i s mt r i p l e
(x'xr,"n).
If
(x'xr,x4),
and morphism argument x, with
an interpretation
d is true for this
formula,
t h e n d ( x r ) a n d d ( x r ) m u s t b e m o r p h i s m sw i t h d o m a i n d ( x r ) a n d c o d o m a i n d ( x n ) .
If
0 F f
f
if
is a sentence (closed formula) of the languageL(ror),
f
is satisfied
we write
in 0 according to the above interpretations.
It
is clear that a first-order
theory for exact additive
categories could be
axiomatized using sentences of L(tor).
our second logical
llle now introduce
exact relation
strongly
predicate Object*r,
almost
categories.
Let r*,
6.2, Definitions.
for
language, apPropriate
type having a unary
denote the structure
binary predicates equality (=) and Unit*r,
ternary
and
predicates Morphi"rRc, Zero*r, SmaIlest*r, Largest*r, Leftproper*,
Rightprop€rR., predicates Negative*r,
four,
predicates Sum*r, Meet*r, Join*,
predicate Compositionr, of arity
a first-order
six.
Converse*, and InclusionRC of arity
and Exact'c of arity
Let L(r*r)
five,
and a
denote the language of
predicate calculus with equality with atomic predicates of
type r*r.
Note that
each atomic predicate
predicate in L(r*r)
L(ror),
( O b j e c t o , c o r r e s p o n d i n gt o O b j e c t * r , e t c . ) .
L(rnc) is a purely relational
the only terms.
structure, with variables of X
Each atomic formula for L(r*r)
e is an almost strongly
variable
Like
is obtained by providing
of X as arguments for one of the seventeen atomic predicates.
variables
If
in L(rlc ) has a corresponding atomic
may represent either
Using the set of variables
d:X+0bjects(0)
is a true
exact relation
category, then each
an object or a morphism of 0, but not both.
X, a function
uMorphisms(0) (aisjoint
interpretation
specifications below.
union)
for each atomic formula of L(r*r)
Again, A, B, C, f,
according to the
g, h, x and y are in X.
x = I
d(x) = 41t; (anY variables)
Object*r(A)
d(A) is an object of 0
Morphism*c(f,A,B)
d(f)
i s a m o r p h i s mf r o m d ( A ) i n t o
d(B) in I
Unit*r(f,A)
d(f) = 1u(o) in e
C o m p o s i t i o n * . ( ,f g , h , A , B , C )
d(f ) = d(g)d(h) f or d(g):d(A)-d(B)
a n d d ( h ) : d ( B) - d (
C) i n I
d(f) = d(g)# for d(g):d(A)..-_d(B)
Converse*r(f,e,A,B)
inB
d(f) s d(g) for d(f) and d(g)
Inclusionnc(f,B,A,B)
from d(A) to d(B) in 0
d(f) = d(g)nd(h) for d(g) and d(h)
Meet*r(f,g,h,A,B)
from d(A) into d(B) in 0
d(f ) = d(e) v d(h) for d(g) and d(h)
Join*r(f ,g,h,^A,B)
from d(A) into d(B) in 0
SmaIIest"c(f,A,B)
d(f) = 00, for D = d(A) and E = d(B) in e
Largestrr(f,A,B)
d(f) = I*
S u m * r ( f, g , h , A , B )
d(f ) = d(g) + d(h) (relational sum)
for D = d(A) and E = d(B) in I
for d(e),d(h):d(A)-613;
in 0
d(f) = -d(g) (relational negative)
Negative*r(f,g,A,B)
f or d(e):d(A)-d(B)
in 0
Zero*r(f,A,B)
d(f) = 00, for D = d(A) and E = d(B) in 0
E x a c t o r ( f, g , A , B , c )
d(f ):d(A)-d(B)
and d(g):d(B)-419;
satisfy d(f )#Id(f ) = d(s)0d(g)# in e
Leftproperrr(f ,A,B)
d(f ):d(A)-d(B)
a n d d ( f ) d (f ) # > l r r o l
in e
Rightproperrr(f,A,B)
d(f):d(A)-d(B)
and d(f)#d(f) = lorrl
in I
Note that d(f ):d(A)-d(B)
is a proper map of I
iff
Leftproper*r(f ,A,B) and
Rightproperrr(f,A,B) are satisfied for d.
As before, arguments labelled A, B or C above are called object arguments,
and arguments labelIed
atomic predicate
its
f,
g or h are called morphism arguments.
Again, each
contains domain and codomain object arguments for each of
morphism arguments, so that
each morphism argument has a norphism triple
as in 6.1.
If
f is a sentencefor L(r*r),
then I ts f
if
f
is satisfied
in I for all
the interpretations
strongly
above.
exact relation
Again, a first-order
theory for almost
categories could be formalized using sentences
in the languageL(rr, ).
6.3. Definitions.
we consider öasic universal
language L(r),
logical
For r a structure type with associated first-order
Horn sentences f having the special
form
(V"r,x2,...,x,r)(l|l),
where W is called the open forrnula for f.
formula PO, or an implication
Pr^PzA...AP.
Here, l{ must either
be an atomic
Q + PO for Q an (open) conjunctive
formula
( m > 1 ) , w h e r eP . = P r ( t 1 , X a , . . . , * r , ) i s a n a t o m i c f o r m u l a
of L(r) for i = 0,1,.
,ß.
(By convention, take m = 0 whenlll equals Po.)
As before, Iet each of tlo., U*r,ür,
u n i v e r s a l H o r n s e n t e n c e sf o r L ( r ) ,
llo and tl, denote the set of basic
where r is to'
TRc, TL, TA or rB,
respective 1y.
Given a ring R with uni.t, we can restate previous definitions
as follows:
u A c ( R ) = { r e t l o r : R - M o dF f } ,
u R c ( R )= { f e U * r : R - R e I F r } ,
u L ( R )= { f e t l r : J ( R ) F r } ,
u A ( R )= { f e U o : o ( R ) F f } ,
U B( R ) = { f e l l , : ß ( R ) F l } .
T h a t i s , e a c h s e t a b o v e c o n t a i n s t h e b a s i c u n i v e r s a l H o r n s e n t e n c e sw h i c h a r e
s a t i s f i e d f o r t h e c o r r e s p o n d i n ge x t e r n a l t h e o r y o f R - m o d u l e s .
W en o w s t a t e o u r t r a n s l a t i o n t h e o r e m f o r b a s i c u n i v e r s a l H o r n s e n t e n c e s .
6 . 4 . T h e o r e m . T h e r e e x i s t r e c u r s i v e f u n c t i o n s T o: t l o - t l '
Tr:Ur*llo'
T r : U o r - - - - . - l l ra,n d T n : U * r - l l o
T, :Ur-U,
suchthat 6'4a'b'c'd'e are
satisfied for alI nontrivial rings R with unit.
6 . 4 a . f e t l o ( R ) i f f T 0 ( f ) e U B ( R )f o r a l l f i n t l o .
6 . 4 b . f e t l r ( R ) i f f T 1 ( f ) e U L ( R )f o r a l l f i n U '
,
in ll'
T a ( f ) e U A c ( R )f o r a I I f
6.4c.
f e ür(R) iff
6.4d.
f e tlor(R) iff
T s ( f ) € U R . ( R ) f o r a I I f i n t l o r'
6.4e.
f e tl*r(R) iff
T4(f ) € UA(R)for all
f in U*, '
We have already discussed the translation
4.10) and T, (see 4.8).
functions To ( inclusion;
T, is based on standard methodsfor
The translation
operations by exactness
r e p r e s e n t i n g s u b o b j e c t s b y m o n o m o r p h i s m sa n d I a t t i c e
we use the
For T'
conditions in an abelian category (see [GH,$3]).
identification
o f R - M o dw i t h P ( R - R e l ) a s i n 5 . 1 2 c . F i n a l l y ,
by adapting 3.16 and the construction of 0 in 5.8 and 5'9.
additional
T, and Tn in the following;
see
T4 is obtained
Wedescribe T,
analysis of the translation
functions
is given in Appendix E.
consider short exact sequences in R-Mod as below:
To construct Tr:Ur-llo'
fr,,
fr,,
.
^
^z-"h-"t---h+
fr,r*,'
fr*t,,
^
''
1-
where h is an even number, h > 4, and A, and A, represent zero objects'
represents an R-module M, then fr,1
an element of su(M), and fr,h*1
same element of su(M).
lattice
If
A1
r e p r e s e n t s a m o n o m o r p h i s mw h o s e i m a g e i s
r e p r e s e n t s a n e p i m o r p h i s mw h o s e k e r n e l i s t h e
In order to define Tr(f)
for f in ll'
we must make
polynomials occurring in f correspond to short exact sequences as above.
These interconnected short exact sequences form a commutative diagram in R-Mod,
in the strong sense that
there
is a most one diagram morphism f . , j'Ar...-----Aj
for each pair (A.,Ar) of diagramobjects, with f.,'
Supposef
i s ( V x , , x 2 , . ' , X r , ) ( l | l ) , 1 l la n o p e n f o r m u l a .
(P, = 9r ^ Pa - {e ^ "'
polynomials p. = Pi(x!,xz
0 < i <m.
(If m= 0, thenW t" po = {0.)
by recursively constructing from f a list
{Po,P,
= ai(x!,xz,...,xrr),
P.}u{{e,{,
F o r s o m e m 2 Q , V r Ie q u a l s
^ P,' = {,n) - Pg = 9g,
for lattice
polynomials,ri
always the unit for A"
l l J eb e g i n t h e d e f i n i t i o n o f T z ( f )
r't2,...,r"
suchthat
q.} q {"r,"2,"',r"},
(s > n) of lattice
ri
is the variable x. for
and k, ! 3 j,k
( i,
i S n, and for each i,
such that r.
is r. Ark or is r. vrk.
f, i+z.t
in the format above (monomorphism
s y m b o l sA . ^ r d f j , u
l{e intend to make
correspond to a short exact sequencewith h = 2i + 2
polynomial r.
each Lattice
I ( i S s, there exist j
Suppose
a n d e p i m o r p h i s mf , , r r * r ) .
> 1) represent systematically chosendistinct
(i,i,k
variables in X = {*r,*a,...}.
The open formula W, for Ta(f) has the form:
U,nU, A ... ^U.nV, nV, n ... ^V. - Uo,
omitted if
with terms U'Ur,...,U.
equation r.
= rk,
1 < i,k
m = 0.
For R-modulesin our context, the
s s, is equivalent to the exactness of
tr,zk+3
'ri+2,L
^
^ z 5 * e + A t + A z k ^* a '
A
For 0 .
i 3 m, theref ore, J.et U. denote the atomic f ormula:
E x a c t o (, f r i * r , r , ' f r , r u * r ' A r 5 * r ' A 1 ' A r u * r ) '
where j
is the smallest integer such that r.
such that rk = {i.
above. )
= pi and k is the smallest integer
(Note that U. corresponds to p. = gi by the discussion
V, describe the commutative diagram of
The hypotheses V'V,
interconnected short exact sequences. First,
let
V'
V'
V, and Vn be
U n i t o r ( f r . r , A 2 ) , Z e r o o r ( f r , r , A 2 , A 2 ) ,U n i t o r ( f , , r , A r ) a n d Z e r o o r ( f r , r , A 3 , A 3 ) ,
respectively.
These four conditions force A, and A, to be zero objects.
Next,
for i = 7,2
E x a c t o (, f r , r r * r ,
f
E x a c t o (, f r r * r , r '
fr,rr*r'
rr*r,r,
E x a c t o r ( f r , r r * r 'f r r * r , r '
Finally,
Ay
Arrty
Arr*r' A1' Arr*r)'
A1 Ar,*r' Ar)'
we must add morphism variables
c o r r e s p o n d i n g t o t h e s e q u e n c eo f l a t t i c e
1 s i s n, ri
n(
iSs,
is just
A,),
a variabl€ "i,
and conditions to the diagram
polynomials rr,rz,..
and nothing further
For
,r".
is needed. Suppose
s o r . i s r . A r k o r " j u . u f o r s o m ej a n d k , 1 s i , k <
i.
we use the diagrams below, with a = 2i +2, b = 2j + 2 and c = 2k+ 2.
Then
f"''
f t'o
A"
,A,
l o
Iro,'
-r
0n the left,
atd f",r,
a s s u m et h a t r . ,
respectively,
fo,rfr,"*1
A"*1
A
- - .b + -1+ A
*t
-*,-'c
1 ,c + 1
c,7
1 ," * 1
lr,r*rl
-;r-
Ä"
f
,A.
l'
A"
r,
tf b + 1 ,
then correspond to r. Aru.
a+1
and ro correspond to the images of f.,1,
fo,,
which is required to be a kernel of
and introduce f.,,
= f.,rfr,r.
and to satisfy f.,1
--a+1
For R*modulesw
, e see that r. will
This can be expressed in the logical
Ianguage
L(rlc ) by the three conditions beLow:
c o m p o s i t i o n o( fr 1 , " * t ,
A'
fr, r, f1,
"+1,
A"*t),
A'
C o m p o s i t i o n o( ,f " , r , f " , o , f o , r , A . , A o , A , ) ,
Exactor(f.,r, fb,c+1,4", Ao, A"*r).
Dually on the right,
r.
= tj t"u
if
r.,
r., and rU correspond to the kernels of
f 1 , " * 1 , f r , r * l . a n d f t , " * t ' r e s p e c t i v e l y , f b * 1 , . * 1 i s a c o k e r n e lo f f " , r f 1 , b + 1 '
and fr, a+1 = fr, r*rfr*r, a+1'
This yields join conditions:
c o m p o s i t i o n o r ( f c , b + 1f ," , 1 , f r , o * r ,
c o m p o s i t i o n o r ( f 1" +
, 1, fr, r*r,
A", A1 Ao*r)'
fb*1,.*1, A'
Ar*r, A"*r)'
E x a c t o r ( f " , b * t ' f b + 1 , " + 1 'A " ' A r * r ' A . * r ) '
The complete list
V'Y2,...,V.
consists of the 4 + 6s - 3n atomic f ormulas
g i v e n a b o v e . S o , W , h a s b e e n d e fi n e d , a n d T r ( f ) i s ( V x ' x a , . . . , X . ) ( W a ) ,
with u chosen minimal so that T2(f)
is closed.
This completes our
description of Tr.
The translation
additivb
relation
T r : U o r . - - - - - - t l af,r o m e x a c t a d d i t i v e
category formulas to
category formulas is straightforward.
SupposeÄ is
(Vxr,"e,,,.xr,)(lll) in llo, with open formula Il equal to
P , n P , l r . . . A P m+ P o ,
f o r a t o m i c f o r m u l a sP o , P r , . . . , P . o f L ( r o r ) .
formula W, for T3(^) will
have the form
U , n U , A , . . A U , , ' n V , n V , n . . . n V r ta U o .
(If m = 0, W is Po.)
Theopen
,1n
(If
Recall that each atomic predicate for L(tor)
m = t = 0, then W, is Uo.)
Define U. to be the atomic
has a corresponding atomic predicate for L(rrr).
for
w i t h t h e s a m ea r g u m e n t l i s t ,
formula corresponding to P'
i = 0,1,.'.,h.
( F o r e x a m p l e ,U 0 i s S u m * r ( " r , " 5 , x 2 , x g , x g ) i f P 0 i s S u m o r ( x ' x u , X z , X B , x n ) . )
V r . c o r r e s p o n dt o t h e l i s t
The terms V'Va
argumentsappearing in the tist
f2,...,fr
of morphism
Suppose
of atomic formulas.
P0,P1,...,P,
t h a t ( f u , A k , B k ) i s t h e m o r p h i s mt r i p l e
f'
corresponding to fu, k < t.
Then
Vr,.-, is LeftproperRc(fk,Ak,Bu)and Vru is Rightproper*c(f'Al,Bk),
F o r e x a m p l e ,i f P o i s E x a c t o c ( " z , X g , X 3 , x n , x g t)h, e r e a r e
k = 1,2,,..,L.
and ("8,*q,ts).
(x'x'xn)
four atomic formulas from the two triples
P, to P. similarly, we obtain Vr.,Vr,..',V2, (t > 0)'
Treating
This defines the open
Note that T3(^) is a
and Tr(Ä) is then (V*r,x2,...,*,)(Wr).
formula W'
for
closed sentence, hence is in U*r, since no new variables were introduced.
As previously noted, its
This completes our description of T,.
properties
are based on the isomorphism between R-Mod and P(R-Rel).
T o d e f i n e T a : U * r - - - - * U o ,w e f i r s t
are called superficial
for alt
recursiveJ.y which case ho1ds.
T' n ( a =) j
certain
sentences. Each superficial
R-ReI or is false for all
t'
identify
= tz)
L (Vx'xr)(x,
sentence is either true
R-Rel, R nontrivial,
For A superficial,
(Vx, )(x, = xr ) if
sentences A of 1l*, which
A is true for all
if
and we can determine
define Tn(A) in Uo by:
R-Rel'
A is false for all R-ReI.
S u p e r f i c i a l s e n t e n c e sa r e o f s i x t y p e s .
Po an atomic formula is called superfieial
AttyA of form (Vx'x2,...,*r)(PO)
of type 7.
Note that
false for aII R-Rel except when Po is of the form x. = tj.
such a Ä is
Hereafter, suppose
A is of form (Vx'x2,..
, X r , ) ( Q- P o ) , w h e r e Q e q u a l s P r n P r n . . . n P . ,
formulas P0,Pr,...,P..
ThenÄ is superficial
variable
of type 2 if
for
for atomic
there is a
x. such that d(*. ) must be both an object and a morphism under any
J
interpretation
-
d satisfying
QAPo.
Such a sentence is essentially
meaningless
for us, because of the requirement that no element of R-ReI be both an object
and a morphism.
Calculation
shows that A is then false
for aII
R-Rel or
t1,
vacuously true for all. R-ReI.
vacuously true
For example, the following
of type 2 because of x, and xr:
sentence is superficial
( V x r ,t z , . . . , * s ) ( ( N e g a t i v " n c ( * r , X s , X o , " r )^ X B = X s ) + U n i t * c ( x 4 , * u ) ) .
of type 3 if
Excluding types 1 and 2, say that A is superficial
and is superficial
form Object*r(xi),
of P'P2,...,P.,
together with consideration of terms P. which are
equations, shows that Tn(A) can be recursively
sentences of these two types.
superficiai
Po has the form
S o m ec a l c u L a t i o n w i t h o b j e c t a r g u m e n t s a n d m o r p h i s m
Morphismrr(xi,xj,ru).
triples
of type 4 if
P0 has the
of type 5 if
computed for
superficial
Excluding types t lo 4, sal that A is
there is an object variable
x. for Po such that
( V x ' x z , . . . , " . ) ( Q = O b i e c t * , ( x ,) ) ,
which is superficial
of type 3, is false for aII R-Rel.
types 1 to 5, A is superfieial
(*r,*j,xu)
of type 6 if
FinalIy,
excluding
there is a morphism triple
for Po such that
(Vx' x?,. ..,*,)(Q = Morphism*r("r,*j,xu)),
which is superficial
superficial
sentence of type 5 or 6 is false
I|le are finally
superficial.
of type 4, is false for all
R-ReI.
in all
0bviously, each
R-ReL
prepared to define T4(A) in the general case, when A is not
Based on 3.16 and 5.Bff,
we will
define below an equation of
rA-polynomials corresponding to each atomic formula P for L(r*r),
primary equation for P.
Also, P may have additional
of ro-polynomials that are ca1led structural
corresponding equations
equations.
There is one structural
equation A = AA# corresponding to each object argument A for P.
:
objects of Y correspond to symmetric idempotents of the additive
algebra Y,)
There are two structural
c o r r e s p o n d i n g t o e a c h m o r p h i s mt r i p l e
calIed the
(Category
relation
equations, f = Af and f = fB,
(f,A,B)
for P.
( T h e H o ms e t i n i
corresponding to symmetric idempotents A and B in Y consists of triples
(A,f,B) with Af = f = fB.)
The primary rA-polynomial equations corresponding
to each P are shown below, together with the number of associated structural
t2
equations.
(Recal1 that p(x) denotes (x+(-x))#(t+(-x)),
(x+(-x))("+(-"))#,
and both x < y and y > x denote x = x^y.)
x=y
X=I,nostructuralequations
Objectra (A)
. A ,= A , 1 s t r u c t u r a l
Morphism*r(f,A,B)
f=f,2structuralequations
Unit*r(f,A)
f=A,Sstructuralequations
Composition*r(f,B,h,A,B,C)
f = Bh, 9 structural equations
Converserr(f,g,A,B)
f=g#,6structuralequations
InclusionRc(f,g,A,B)
fsg,6structuralequations
Meet*r(f ,g,h,A,B)
f = gAh,
B structural equations
Join*r(f,g,h,A,B)
f - gvh,
B structural equations
Smallest*c(f,A,B)
f=P(A)p(B),4structuralequations
Largest*r(f,A,B)
f={(A)q(B),4structuralequations
S**r(f
f - g+h,
,g,h,A,B)
equation
B structural equations
Negative*r(f,g,A,B)
f = -g, 6 structural
Zero*r(f,A,B)
f = q(A)p(B), 4 structural
Exact*r(f,g,A,B,C)
f#q(l)f
Leftproper*r(f,A,B)
ff# > A, 4 structural
equations
R i g h t p r o p e r r ,( f , A , B )
f#f . B, 4 structural
equations
For example, Zero*r("s,*5,xr)
the structural
Note that
in i
q(x) denotes
equations
equations
= gp(B)g#, 7 structural
equations
has the primary equation x3 = i[(xs)p(*e) and
equations xu = xuxu#, x, = xrxr#, x3 = x5xg and xr' = XgX6.
the primary equations correspond to the morphism graph formulas
to" Y an additive
relation
For A not superficial,
al.gebra.
the open formula 1{, for Tn(Ä) has the form
U ,n U , A . . . A U . n V ,n V , n . . . n V . * U o .
Here, U. is the primary equation corresponding to P., for
The list
V.Yz,,..,Vt
i = 0,!,
(t > 0) is obtained by aggregating all
,fr.
the structural
13
f o r m u l a s c o r r e s p o n d i n gt o t h e a t o m i c f o r m u l a s P ' P r , . ' , , P r .
Fina1ly, T4(A)
i s ( V x ' x 2 , . . . , X r , ) ( W l ) , w h i c h i s a c l o s e d s e n t e n c e ,h e n c e i s i n i n U o ,
because no new variables were introduced.
The atomic formula Po of Ä for i
primary equation and its
structural
"o""""nonds to a conjunction of its
If
equations.
the
^ is not superficial,
structural
e q u a t i o n s f o r P o a r e c o n s e q u e n c e so f Q , b e c a u s e t y p e s 5 a n d 6 a r e
excluded.
In that case, PO is true iff
see why this
its primary equation is true.
To
approach is necessary, suppose A is
( V x r , * a , " 3 ) ( L e f t p r o p e t R c ( t r , * a , * , ) - L e f t p r o p e " n c( " r , x z , x z ) ) '
This sentence is superficial
of type 6, hence false for all
computed by the procedure for non-superficial
trivially
R-Rel.
If
sentences, Tn(A) would be a
true sentence, with both Uo and U, equal to xrxru = "r.
This completes the description of Tn, closing the loop of recursive
translation
funct ions .
we
L C M 7 V 4_ T O / 3 I / B B
57. Equivalence and Inclusion
In the introduction,
of Rings for Module Theories.
we asserted that
our five
theory for universal
determined a unified
to any ring.
preserved relative
the unified
for modules
ltle have showedthat basic
sentences.
between these theories,
Horn sentences can be translated
universal
external theories
with truth
We can compare two rings R and S according to
theory: we say that R is srnaller than S if
it
has fewer models,
say ß(R) c ß(S), or more true universaL basic Horn sentences, SaYUB(R) : Ur(S).
( It
rings minimal, and the ring Z of integers
to make trivial
seemsnatural
maximal; Z corresponds to the most general module theory,
groups.)
In the following,
that
show that these ring comparisons can be
we wiIl
characterized by the existence of exact embeddingfunctors.
abelian category techniques available
7.7. Definitions
and Properties.
diagram-smaller than S, written
If
R-llod+S-llod.
functor
to S, and write R - S.
relation
7.Ia.
-
This makes many
for analysis of such comparisons.
For rings R and S with unit,
R J S, if
there exists
R J S and S J R, say that
The relation
say that R is
an exact embedding
R is diagram-equivalent
J is calted diagram inclusion,
and the
is called diagram equivafence.
relation
Diagram inclusion is a symmetric and transitive
on the class of alI
the class of all
It
of abelian
rings.
(quasiorder)
Diagramequivalence is an equivalence relation
on
rings.
is convenient to state the unification
results
at this
point,
although
part of the proof must be deferred,
7 . 2 . T h e o r e m . S u p p o s eR a n d S a r e r i n g s w i t h u n i t .
all
Then the following are
equivalent:
7.2a.
R J S, that
7.2b.
There exists
7.2c.
J(R) s f(S).
is,
there exists
an exact embeddingfunctor R-Mod+S-Mod.
an embeddingrelation
(Every lattice
functor R-Rel
'S-ReI.
w h i c h i s r e p r e s e n t a b l e b y a n R - m o d u l ei s
r e p r e s e n t a b l e b y a n S - m o d u l e .)
7.2d.
0(R) S 0(S).
(Every additive relation algebra which is representable
by an R-module is representable
7.2e.
by an S-module' )
(Every additive relation algebra with unit which is
ß(R) c ß(S).
r e p r e s e n t a b l e b y a n R - m o d u l e i s r e p r e s e n t a b l e b y a n S - m o d u l e .)
7.2f.
UAc(R): tlor(S).
is satisfied
7,2g.
7.2h.
for S-Modis satisfied
URc(R): ll*r(S).
is satisfied
f o r R - l l o d .)
(Every basic universal Horn sentenceof type rRc which
for S-ReI is satisfied
for R-ReL )
(Every basic universal Horn sentence of type r, which
UL(R) : tlr(S).
is satisfied
(Every basic universal Horn sentenceof type ro, which
in every Su(M), M an R-module, is satisfied
in every Su(N), N an
S - m o d u l e .)
7.2i.
UA(R): tlo(S).
is satisfied
(Every basic universal Horn sentenceof type ro which
in every Su(MeM), M an R-module,is satisfied
in every Su(N@N),
N a n S - m o d u l e .)
7.2i.
UB(R) 2 UB(S).
is satisfied
(Every basic universal Horn sentence of type r, which
in every Su(MeM), M an R-module,is satisfied
in every Su(N@N),
N a n S - m o d u l e .)
We have already proved many parts of this
We know that
theorem.
are equivalent by 5.18, and that 7.2c,d,e are equivalenL by 4.It.
l o o p o f t r a n s l a t i o n f u n c t i o n s ( T h e o r e m6 . 4 ) , i t
are all
by 5.6b.
equivalent.
If
0(R) = 0(S) iff
U L ( R )= U L ( S ) i f f
R and S are rings with unit,
functors R-ReI+S-Rel
embeddingrelation
iff
Also, 7.2b - 7.2e
section, to complete the proof
immediate consequences of Theorem 7 .2 nexL '
l{e give several
7.3. Corollary.
From the
follows that 7.2f ,g,h,i,i
Clearly 7.2c,h are equivalent by 2.9.
show 7.2e + 7.2a laLer in this
l{e will
7.Za,b
ß(n) = 615; iff
U A ( R )= U A ( S ) i f f
then R - S iff
and 5-p"1+R-ReI
U A c ( R )= U A c ( S ) i f f
iff
there exist
8(R) = 9151
U R c ( R )= t l * r ( S ) i f f
U B ( R )= U B ( S ) '
7.4. Corollary.
Each ring is diagram-equivalent to somecountable ring (2.I0).
7.5. Corollary.
S u p p o s eR a n d S a r e r i n g s w i t h u n i t ,
and f
is a (closed,
p r e n e x ) u n i v e r s a l s e n t e n c e i n o n e o f t h e l a n g u a g e sL ( t o r ) , L ( r * , ) ,
L(rr),
If
L(ro) or L(rr).
for R-modules(that
f is satisfied
is,
throughout R-Mod,
then there exist basic universal
R-Rel, J(R), 0(R) or ß(R), respectively),
Horn sentences fr,l,
f,
is satisfied
^l'
i s a l o g i c a l c o n s e q u e n c eo f l r n f a n . . .
such that f
for S-modules, then it
is satisfied
If R J S and f
for R-modu]es.
Since J(R), 0(R) and ß(R) are quasivarieties by 2.9 and 4.9, 7.5 follows
immediatelyfor f in .8(rr), I(to)
S e e A p p e n d i xE f o r f
or f(rr).
in J(ru, )
or J(tnc ).
lle will
prove Theorem7.2 using the methods of Fuller
which exploit
and Hutchinson [*x],
t h e s p e c i a l p r o p e r t i e s o f e n d o m o r p h i s mr i n g s E n d ( * R ( F ) ) , w h e r e
B is a sufficiently
large infinite
number.
cardinal
An abbreviated treatment
is given below; consult [xx] for the motivating ideas from the theory of left
coherent rings.
7.6. Definitions
and Properties.
Let R be a ring,
Iet p be a cardinal
n u m b e r w i t h B > x o + l R l , a n d l e t T d e n o t e t h e e n d o m o r p h i s mr i n g E n a ( r R ( B ) ) .
Let B be a set of free generators for *R(p), with
lBl = F.
I f M i s a s u b m o d u l eo f R R ( P ) , t h e r e i s a n e n d o m o r p h i s mt i n T s u c h
t h a t I m t = M ( s i n c e l * R ( p ) l - p b e c a u s ep > x o + l R l , a n d * R ( F ) i s f r e e o n p
7.6a.
generators ) .
7.6b.
For t and u in T, u is in Tt iff
7.6c. The principal
lattice
left
Su(rT) of left
ideals {Tt:
7.6d.
If
t e T} of T form a sublattice
ideals of T, and this
Su(*R(F); via the mapTtllli-Imt
to the lattice
Im u s Im t.
of lattice
(7.6a,b).
sublattice
Similarly,
of the
is isomorphic to
Su(rT) is isomorphic
p
ideals of Su(RR( ) ).
P = RR(B), then for each n > 1, P = P(n) in R-Mod. (Partition
B into subsets B'Br,...,Bn,
each of cardinality
s u b m o d u l eo f P g e n e r a t e d b y B i .
sum for P such that P = Pi for
7.7. Proposition.
ThenPr*Prt...
F, and let P. denote the
*Pn is an internal direct
each i s n. )
If R is a ring with unit and p > 1, then R - End(nR(F)).
S u p p o s eB i s a f r e e g e n e r a t i n g s e t f o r * R ( F ) a n d T = E n d ( R R ( p ) ) .
Note that *R,lp) is a bimodule and RR(p) is a projective generator. So, G =
and R J T. Also,
H o m * ( * R [ F) , - ) i s a n e x a c t e m b e d d i n g f u n c t o r R - l l o d - T - M o d ,
Proof:
= rb for r in R and b in B is a ring homomorphism
given by p(r)(b)
g:R-T
preserving 1.
S o , g i n d u c e s a n e x a c t e m b e d d i n gf u n c t o r
This proves R - T.
c h a n g eo f r i n g s ( r v = q ( r ) v f o r v i n r M a n d r i n R ) .
S o m es p e c i a l d e f i n i t i o n s
7.8. Definitions
Hn:T-Mod-R-Mod by
are given next.
and Properties.
h o m o m o r p h i s mr | : E n d ( n M ) - E n d ( s N )
F o r M i n R - M o da n d N i n S - M o d , a r i n g
is said to preserve exaetness if
g in End(*M) such that (f ,g) is exact in R-llod, (Vtf ),rl(g))
We say that rf reflects
I
epimorphisms if
whenever {(f)
f or f and
is exact in S-Mod.
is an epimorphism (onto),
then f must be an epimorphism.
7.9. Definitions
and Properties.
for any lr,lr,...,t'
Ur=, "iti
A right
T-moduleN, is called 1-flat
in T (n > 1), there exist 51,sa,...i".
= 0 , a n d f o r a n y v 1, Y z , ' . ' , v ,
in T suchthat
i n N s u c h t h a t [ J . = 1v i t i
exists u in N such that us. = vi for i < n.
if
= 0, there
Note that s1,"2t...,sr,
depend
only upont'tr,...,trr,
and not on any particular choice of ,!,Y2,...,vn,
7.9a.
then it
If NT is l-flat,
is flat.
We are now prepared to complete the proof of Theorem7.2, and also
obtain two more technical
characterizations
of diagram inclusion.
7 . 1 0 . T h e o r e m . S u p p o s eR a n d S a r e r i n g s w i t h u n i t ,
s o m ec a r d i n a l .p > x o + l n l .
7.I}a.
t
S iff
:For some S-module N, there exists
{:T-End(
7.10b.
T h e nR J
and T = End(RR(p)) for
ß(R) gß(S)
iff
7.10a iff
a (unit-preserving)
7.t1b.
ring monomorphism
preserves exactness and ref lects epimorphisms.
rN) which
There exists a bimodul" sNr such that N, is 1-flat
and Nt + N for each
in T which is not an epimorphism.
Proof: As previously noted, R J S implies ß(R) S ß(S) by 5.18 and 5.6b.
A s s u m eß ( R ) g ß ( S ) , s o t h e r e e x i s t s a r r - e m b e d d i n g [ : R e l ( r R t F ) ; * R " 1 1 r N 1
for
some S-module N.
By 3.16i,
t h e r i n g o f e n d o m o r p h i s m so f a n y m o d u l e M i s
the ring of proper maps in ReI(M).
isomorphic to hom(!,!),
proper morphisms produces an rf satisfying
Restricting
t to
This proves that ß(R) g ß(S)
7.10a.
implies 7.1.0a.
A s s u m et h a t r f : T - E n d (
certain
function
rN)
is convenient to write
It
satisf ies 7 .70a.
symbol and reverse order:
evaluations using a binary infix
A bimodule structure rN, can be defined using vt = v*rJi(t)
are in T, n > 1. Since *P = *P(")
f o r v e N a n d t e T . S u p p o s et ' L r , . . . , t '
b y 7 . 6 d i f , P d e n o t e " * R ( F) , w e c a n s e l e c t i n s e r t i o n s t c . a n d p r o j e c t i o n s n .
in T for i s n such that rc.n. = 1 for i s n, trtj = 0 for i + j, t 3 i,i = n,
denotes f(x).
xxf
n
n
and [J.=r Tjrcj - 7.
Let t = Uj=, tjtj,
thereexists s inT
suchthat Ims=Kert.
n
Ur=, "rti
n
= t.
so that r.t
for i < n'
D e f i n es i = s f i i f o r i < n .
So,
n
= Ur=1sri*rt
= "(Ui=r *iK, )t = s1t = 0'
= 0 f or v, ,y2,...,v'
N o ws u p p o " "U : = , t i t i
Let v = U;=1 ViKi, so
in N.
v * , . 1 ( t ) = ( U l = 1 v i * , ' 1 ( r i) ) * . r | ( t ) = U l = 1 v i * 9 ( t c i t ) = U l = r v i t i
Since rf preserves exactness, Ir,rl(s)
such that u * {(s)
By 7'6a,
= v.
= Ker t(t),
= 0'
and so there exists u in N
But then
n
. (uxrl("))xV(ni)= u*,|(ri,
r"i = usrT=
t U . i = lv j * , i ( r j ) ) * r l ( n . ) = v , ,
for i = !,2,...,D.
Therefore,N, is 1-f lat.
is not onto by hypothesis, so Nt = Im,l(t)
A s s u m e7 . 1 0 b , a n d l e t
If
+ N.
t
in T is not onto, then rJi(t)
Therefore, 7.t}a
implies 7.10b'
F denote the composite functor
R-Mod+T-Mod+s-Mod,
where G is Hom(**|U,,-) and H is rNer-.
e x a c t b o c a u s eN , i s f l a t
(7.9a), F is exact a1so.
Let K be a proper left
RP,
and bo is in B,
set B-{bo}.
iff
ideal of R.
SupposeB is a free generating set for
Let Ko = KbovPo in Su(*P), where Po is generated by the
By 7.6a, there exists t
in T such that Im t = Ko.
Since w e Tt
\:T/TI-G(R/K),
Im w s Im t by 7.6b, there is a T-linear monomorphism
given by \(1+Tt)
aII
Since G is exact by 7.7 and H is
b in B-{bo}.
= \ o i n H o m ( R P , R R / Kw) ,h e r e \ 0 ( b 0 ) = 1 + K a n d \ 0 ( b ) = 0 f o r
Since t
is not onto, NTt = Nt + N by hypothesis, and so
sNsr
(T/Tt) = N/NTI + 0.
But then
F ( R / K ) = H ( G ( R / K ) )= , l l s T G ( R / K )+ 0 ,
v i a t h e m o n o m o r p h i " tN S r \ .
7 . 1 0 b i m p l i e sR J S .
I
T h e n F i s a n e x a c t e m b e d d i n gf u n c t o r , a n d s o