Introduction
Some preliminaries on clone theory
Results
Conclusion
Reference
Meet-reducible submaximal clones
determined by nontrivial equivalence
relations.
By:
Temgoua Alomo Etienne R.
University of YAOUNDE I; Cameroon
February 10, 2017
DIEKOUAM,TEMGOUA,TONGA
Workshop AAA93; Feb. 2017
Introduction
Some preliminaries on clone theory
Results
Conclusion
Reference
Introduction
Let Ek = {0, 1, . . . , k − 1} be a finite set of k elements with k ≥ 1.
The set of all n-ary
on Ek is denoted by On (Ek ) and we
S operations
set O(Ek ) =
On (Ek ). A clone on Ek is a subset F of O(Ek )
0<n<ω
which contains all the projections and is closed under composition.
The clones on Ek , ordered by inclusion, form a complete lattice
denoted
by L(Ek ) whose least element is
Q
n
= {ei ; n ∈ ω and 1 ≤ i ≤ n} and whose greatest element is
O(Ek ). A clone C ∈ L(Ek ) is called maximal if it is covered only
by O(Ek ). A clone C ∈ L(Ek ) is called submaximal if it is covered
only by a maximal clone.
DIEKOUAM,TEMGOUA,TONGA
Workshop AAA93; Feb. 2017
Introduction
Some preliminaries on clone theory
Results
Conclusion
Reference
Introduction
The set of operations on Ek preserving ρ is a clone denoted by
Pol(ρ). The maximal clones have been described for
k = 2(respectively k = 3 and k ≥ 4) in Post[7](respectively
Jablonskij [4] and Rosenberg [10, 11]). They are of the form
Pol(ρ) where ρ belongs to one of six families of relations which
include some familiar and easily defined relations. All submaximal
clones are known for k = 2 (see [7]) and k = 3 (see [5]).
We characterize here, all relations ρ such that the clones of the
form Pol(θ) ∩ Pol(ρ) is maximal in Pol(θ), where θ is a nontrivial
equivalence relation on a given finite set.
DIEKOUAM,TEMGOUA,TONGA
Workshop AAA93; Feb. 2017
Introduction
Some preliminaries on clone theory
Results
Conclusion
Reference
Definitions and Rosenberg’s theorem
preliminaries
Definition 2.1
Let h be a positive integer.
(i) An h-ary relation ρ is called totally symmetric if for every
permutation σ of {1, . . . , h} and each tuple (a1 , . . . , ah ) ∈ Ekh ,
(a1 , . . . , ah ) ∈ ρ iff (aσ(1) , . . . , aσ(h) ) ∈ ρ.
(ii) τhEk is the h-ary relation defined by (a1 , . . . , ah ) ∈ τhEk iff there
exists i, j ∈ {1, . . . , h} such that i 6= j and ai = aj .
(iii) An h-ary relation ρ is called totally reflexive if τhEk ⊆ ρ.
DIEKOUAM,TEMGOUA,TONGA
Workshop AAA93; Feb. 2017
Introduction
Some preliminaries on clone theory
Results
Conclusion
Reference
Definitions and Rosenberg’s theorem
preliminaries
Definition 2.2
Let h be a positive integer.
(iv) If ρ is totally reflexive and totally symmetric, we define the
center of ρ denoted by Cρ as follows:
Cρ = {a ∈ Ek : (a, a2 , . . . , ah ) ∈ ρ, for all a2 , . . . , ah ∈ Ek }.
(v) A permutation π is prime if all cycles of π have the same
prime length.
(vi) We call a subset σ ⊆ Ek4 affine if there is a binary operation +
on Ek such that (Ek , +) is an abelian group and
(a, b, c, d) ∈ σ ⇔ a + b = c + d holds. An affine relation σ is
prime if (Ek , +) is an abelian p-group for some prime p, that
is, all elements of the group have the same prime order p.
DIEKOUAM,TEMGOUA,TONGA
Workshop AAA93; Feb. 2017
Introduction
Some preliminaries on clone theory
Results
Conclusion
Reference
Definitions and Rosenberg’s theorem
preliminaries
Definition 2.3
Let h be a positive integer.
(vii) For h ≥ 2, an h-ary relation ρ on Ek is called central, if ρ has
the three following properties: ρ is totally reflexive and
non-diagonal (ρ 6= Ekh ); ρ is totally symmetric and Cρ 6= ∅.
(viii) For h ≥ 3, a family T = {V1 ; · · · Vm } of equivalence relations
on Ek is called h-regular if each Vi has exactly h equivalence
classes and ∩{Bi |1 ≤ i ≤ m} is non-empty for arbitrary
equivalence classes Bi of Vi , 1 ≤ i ≤ m.
(ix) For 3 ≤ h ≤ k, a h-regular relation on Ek determined by the
h-regular family T (and often denoted by λT ), consists of all
h-tuples whose set of components meets at most h − 1 classes
of each Vi ( 1 ≤ i ≤ m).
DIEKOUAM,TEMGOUA,TONGA
Workshop AAA93; Feb. 2017
Introduction
Some preliminaries on clone theory
Results
Conclusion
Reference
Definitions and Rosenberg’s theorem
preliminaries
By PolEk (σ) or, briefly, Pol(σ) we denote the set of all functions
f ∈ OEk that preserve the relation σ. For Q ⊆ REk , we put
PolEk (Q) := ∩ PolEk (σ). The set of all relations σ ∈ REk
σ∈Q
preserved by the function f ∈ OEk is denoted by Invk (f ). For
A ⊆ OEk let Invk (A) := ∪ Invk (f ) be the set of all invariants of A
f ∈A
and let Invnk (A) := (Invk (A)) ∩ Rnk be the set of all n-ary invariants
of A. If k can be seen from the context, we write Inv instead of
Invk , Invn instead of Invnk for n ∈ N. Elementary connections
between Pol and Inv are summarized in the following theorem.
DIEKOUAM,TEMGOUA,TONGA
Workshop AAA93; Feb. 2017
Introduction
Some preliminaries on clone theory
Results
Conclusion
Reference
Definitions and Rosenberg’s theorem
preliminaries
Theorem 2.1
For arbitrary A, B ⊆ OEk and arbitrary S, T ⊆ REk , it holds:
(a) A ⊆ B ⇒ Inv(B) ⊆ Inv(A), S ⊆ T ⇒ Pol(T ) ⊆ Pol(S);
(b) A ⊆ Pol Inv(A), S ⊆ Inv Pol(S);
(c) A ⊆ Pol(S) ⇔ S ⊆ Inv(A);
(d) Pol(S ∪ T ) = Pol(S) ∩ Pol(T ), Inv(A ∪ B) = Inv(A) ∩ Inv(B).
(e) Inv Pol Inv(A) = Inv(A), PolInv Pol(S) = Pol(S);
((a) and (b) mean that the pair (Pol, Inv) of mappings
Inv : P(OEk ) → P(REk ), A 7→ Inv(A) and
Pol : P(REk ) → P(OEk ), Q 7→ Pol(Q) is a Galois connection
between the ordered sets (OEk , ⊆) and (REk , ⊆).
DIEKOUAM,TEMGOUA,TONGA
Workshop AAA93; Feb. 2017
Introduction
Some preliminaries on clone theory
Results
Conclusion
Reference
Definitions and Rosenberg’s theorem
preliminaries
By the Theorem 2.1, we can state that:
Theorem 2.2
[5] Let A be a clone of OEk . Then A = Pol Inv(A).
Then we see that the maximal clones are generated only by one
relation. The following theorem due to I. G. Rosenberg characterize
all types of relations such that Pol(ρ) is a maximal clone.
DIEKOUAM,TEMGOUA,TONGA
Workshop AAA93; Feb. 2017
Introduction
Some preliminaries on clone theory
Results
Conclusion
Reference
Definitions and Rosenberg’s theorem
Rosenberg’s theorem
Theorem 2.3
(I. G. Rosenberg [11]). Let 1 < k < ℵ0 . A clone C on Ek is
maximal if and only if it is of the form Pol(σ) , where σ is an h-ary
relation belonging to one of the following classes(called
Rosenberg’s list ( RBL )):
1
The set of all partial orders with least and greatest element
2
The set of all graph of prime permutations
3
The set of all non-trivial equivalence relations
4
The set of all prime-affine relations
5
The set of all central relations
6
The set of all h-regularly generated relations
DIEKOUAM,TEMGOUA,TONGA
Workshop AAA93; Feb. 2017
Introduction
Some preliminaries on clone theory
Results
Conclusion
Reference
Cases where we have not submaximality
Theorem 3.1
If θ is a nontrivial equivalence relation and ρ is a partial order with
least and greatest element, on a finite set Ek , then Pol(θ) ∩ Pol(ρ)
is not submaximal in Pol(θ).
DIEKOUAM,TEMGOUA,TONGA
Workshop AAA93; Feb. 2017
Introduction
Some preliminaries on clone theory
Results
Conclusion
Reference
Cases where we have not submaximality
Theorem 3.2
Let α be a prime affine relation and θ a nontrivial equivalence
relation. We have
Pol(θ) ∩ Pol(α)
Pol(α1 )
Pol(θ),
where
α1 = {(a, b, c, d) ∈ α : (a, b), (a, c), (a, d) ∈ θ}.
DIEKOUAM,TEMGOUA,TONGA
Workshop AAA93; Feb. 2017
Introduction
Some preliminaries on clone theory
Results
Conclusion
Reference
Cases where we have submaximality
Theorem 3.3 ([3])
If θ is a nontrivial equivalence relation and ρ is a graph of prime
permutation π, on a finite set Ek , then Pol(θ) ∩ Pol(ρ) is
submaximal in Pol(θ) if and only if θ and ρ satisfy one of the
following statements:
(a) ρ
θ
(b) The image of an equivalence class of θ is include in another
class of θ surjectively.
DIEKOUAM,TEMGOUA,TONGA
Workshop AAA93; Feb. 2017
Introduction
Some preliminaries on clone theory
Results
Conclusion
Reference
Cases where we have submaximality
Theorem 3.4
If θ and ρ are distinct nontrivial equivalence relations on a finite
set Ek , then Pol(θ) ∩ Pol(ρ) is submaximal in Pol(θ) if and only if
θ and ρ satisfy one of the following statement:
(a) θ
ρ or ρ
θ
(b) ρ ∩ θ = ∆Ek and ρ ◦ θ = ∇Ek
DIEKOUAM,TEMGOUA,TONGA
Workshop AAA93; Feb. 2017
Introduction
Some preliminaries on clone theory
Results
Conclusion
Reference
Cases where we have submaximality
For the next results, we give some definitions to be used. Let ρ be
an h-ary relation (h > 1), and θ be a nontrivial equivalence relation
on Ek with t classes on Ek . For i ∈ {0; 1; 2; · · · ; h − 1} we define
the relation ρi,θ by
n
ρ0,θ = (a1 , . . . , ah ) ∈ Ekh /∃ui ∈ [ai ]θ , i ∈ {1; · · · ; h}
with (u1 , u2 , . . . , uh ) ∈ ρ} .
and for i ≥ 1,
n
ρi,θ = (a1 , . . . , ah ) ∈ Ekh /∃uj ∈ [aj ]θ , j ∈ {i + 1; · · · ; h}
with (a1 , a2 , . . . , ai , ui+1 , . . . , uh ) ∈ ρ} .
DIEKOUAM,TEMGOUA,TONGA
Workshop AAA93; Feb. 2017
Introduction
Some preliminaries on clone theory
Results
Conclusion
Reference
Cases where we have submaximality
We give some definitions to be used. Let ρ be an h-ary relation
(h > 1), and θ be a nontrivial equivalence relation on Ek with t
classes on Ek .
For σ ∈ Sh and γ an h-ary relation, we set
γσ = {(aσ(1) , . . . , aσ(h) )/(a1 , . . . , ah ) ∈ γ}.
Definition 3.1
There is a transversal T for the θ-classes means that there exist
u1 , . . . , ut ∈ Ek such that (ui , uj ) ∈
/ θ for all 1 ≤ i < j ≤ t,
(ui1 , ui2 , . . . , uih ) ∈ ρ for all 1 ≤ i1 , . . . , ih ≤ t and
T = {u1 , . . . , ut }.
DIEKOUAM,TEMGOUA,TONGA
Workshop AAA93; Feb. 2017
Introduction
Some preliminaries on clone theory
Results
Conclusion
Reference
Cases where we have submaximality
Definition 3.2
There is a transversal T of order l (1 ≤ l ≤ h − 1) for the θ-classes
means that there exist u1 , . . . , ut ∈ Ek such that (ui , uj ) ∈
/ θ for all
1 ≤ i < j ≤ t, (a1 , a2 , . . . , al , vl+1 , vl+2 , . . . , vh ) ∈ ρ, for all
a1 , a2 , ..., al ∈ Ek ; vl+1 , vl+2 , . . . , vh ∈ {u1 ; · · · ; ut }; and
T = {u1 , ..., ut }.
A transversal of order 0 is simply a transversal for the θ-classes.
DIEKOUAM,TEMGOUA,TONGA
Workshop AAA93; Feb. 2017
Introduction
Some preliminaries on clone theory
Results
Conclusion
Reference
Cases where we have submaximality
Definition 3.3
1
ρ is θ-closed means that ρ = ρ0,θ .
2
ρ is weakly θ-closed of order l(1 ≤ l ≤ h − 1) means that
there is a transversal
T T = {u1 ; · · · ; ut } of order l − 1 for the
θ-classes and ρ =
(ρl,θ )σ .
σ∈Sh
DIEKOUAM,TEMGOUA,TONGA
Workshop AAA93; Feb. 2017
Introduction
Some preliminaries on clone theory
Results
Conclusion
Reference
Cases where we have submaximality
We obtain the following characterization.
Theorem 3.5
Let k ≥ 3 and let θ be a nontrivial equivalence relation on Ek with
t equivalence classes. For an h-ary central relation ρ on Ek , the
clone Pol(ρ) ∩ Pol(θ) is a submaximal clone of Pol(θ) if and only if
h ≤ t and ρ satisfies one of the following three conditions:
I. η ⊆ ρ and every θ-class contains a central element of ρ;
II. ρ is θ-closed;
III. ρ is weakly θ-closed of order l and η ⊆ ρ.
DIEKOUAM,TEMGOUA,TONGA
Workshop AAA93; Feb. 2017
Introduction
Some preliminaries on clone theory
Results
Conclusion
Reference
Cases where we have submaximality
and this other result.
Theorem 3.6
Let θ be a nontrivial equivalence relation and ρ be an h-regular
relation on Ek determined by the h-regular family
T = {θ1 ; · · · ; θm }. Pol(θ) ∩ Pol(ρ) is maximal in Pol(θ) if and only
if ρ is θ-closed.
DIEKOUAM,TEMGOUA,TONGA
Workshop AAA93; Feb. 2017
Introduction
Some preliminaries on clone theory
Results
Conclusion
Reference
Conclusion
We have characterized the relations ρ from the Rosenberg’s list for
which Pol(θ) ∩ Pol(ρ) is maximal in Pol(θ), where θ is a nontrivial
equivalence relation. The classification of all central relations ρ on
Ek such that the clone Pol(θ) ∩ Pol(ρ) is maximal in Pol(θ)
improves Temgoua and Rosenberg’s results [12]. To complete our
work, we plan to characterize the meet-irreducible submaximal
clones of Pol(θ) for a nontrivial equivalence relation θ.
DIEKOUAM,TEMGOUA,TONGA
Workshop AAA93; Feb. 2017
Introduction
Some preliminaries on clone theory
Results
Conclusion
Reference
K. A. Baker and A. F. Pixley, Polynomial interPolation and the
Chinese remainder theorem for algebraic systems. Math. Z.
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S. Burris and H.P. Sankappanavar, A Course in Universal
Algebra. Springer-Verlag, 1981.
Grecianu, A.: Clones sous-maximaux inf-réductibles. Mémoire
numérisé par la Division de la gestion de documents et des
archives de l’Université de Montréal
http://hdl.handle.net/1866/7887 (2009)
S.V. Jablonskij, Functional constructions in many-valued
logics. Tr. Mat. Inst. Steklova 51, 5-142 (1958) (Russian)
D. Lau, Function algebras on finite sets. Springer, 2006.
DIEKOUAM,TEMGOUA,TONGA
Workshop AAA93; Feb. 2017
Introduction
Some preliminaries on clone theory
Results
Conclusion
Reference
R. Poeschel, Concrete representation of algebriac structure and
a general Galois theory. Contribution to General Algebra (Proc.
Klagenfurt Conf. 1979), Verlag J. Heyn, Klagenfurt; 249 - 272.
E.L. Post Introduction to a general theory of elementary
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E. L. Post The two-valued iterative systems of mathematical
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1941.
R.W. Quackenbush, A new proof of Rosenberg’s primal
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28 (Finite algebra and multiple valued logic). North Holland,
Amsterdam 1971; 603 - 634.
DIEKOUAM,TEMGOUA,TONGA
Workshop AAA93; Feb. 2017
Introduction
Some preliminaries on clone theory
Results
Conclusion
Reference
I.G. Rosenberg, La structure des fonctions de plusieurs
variables sur un ensemble fini. C. R. Acad. Sci. Paris Ser. A-B
260, 3817.3819 (1965) [10]
I.G. Rosenberg, Functional completeness in many-valued
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Vĕd 80, 3-93 (1970) (German)
E. R. A. Temgoua and I. G. Rosenberg Binary central relations
and submaximal clones determined by nontrivial equivalence
relations. Algebra Universalis, Springer Basel AG, 67, 2012,
299-311.
DIEKOUAM,TEMGOUA,TONGA
Workshop AAA93; Feb. 2017
Introduction
Some preliminaries on clone theory
Results
Conclusion
Reference
THANKS FOR YOUR
KINDLY
ATTENTION!
DIEKOUAM,TEMGOUA,TONGA
Workshop AAA93; Feb. 2017