On some sufficient condition for the equalityof multi-clone and super-clone Текст научной статьи по специальности «Математика»

Journal of Siberian Federal University. Mathematics & Physics 2018, 11(1), 97—102

УДК 519.7

On some Sufficient Condition for the Equality of Multi-clone and Super-clone

Nikolay A. Peryazev*

Saint Petersburg State Electrotechnical University Professor Popov, 5, Saint Petersburg, 197376

Russia

Ivan K. SharankhaeV

Institute of Mathematics and Informatics Buryat State University Smolin, 24a, Ulan-Ude, 670000

Russia

Received 05.04.2017, received in revised form 18.07.2017, accepted 28.09.2017 Multi-clones and super-clones are considered in this paper. They are generalizations of clones. To get a super-clone one need to add to a multi-clone the closure condition with respect to solvability of the simplest equation. The condition for identity of multi-clone and super-clone is proved.

Keywords: multi-operation, multi-clone, super-clone, superposition, operation, substitution. DOI: 10.17516/1997-1397-2018-11-1-97-102.

Introduction

Clones are studied most actively in the theory of functional systems [1]. Clones are sets of operations that are closed with respect to superposition, and they contain all projection operators. Recently interest in generalizations of clones, namely, hyperclones, multiclones and superclones has been raised [2].

Multi-clone is a set of multi-operations which are closed with respect to superposition, and it contains all complete, empty and projection operations. A super-clone is obtained from a multi-clone by adding the closure condition with respect to solvability of the simplest equation. It is known that super-clones are closely related to clones. Complete Galois connection between them was established [3]. Condition of the equality of multi-clone and super-clone is obtained in this paper.

Let A be an arbitrary finite set, and B(A) be the set of all subsets of A including 0.

A mapping from An into A is described as an n-ary operation on A (the case n = 0 is possible). The set of all n-ary operations on A is described as PA, and the set of all operations on A is described as

Pa = u Pa-

A mapping from An into B(A) is described as an n-ary multi-operation on A (the case n = 0 is possible). The set of all n-ary multi-operations on A is described as MA, and the set of all

* [email protected] 1 [email protected] © Siberian Federal University. All rights reserved

multi-operations on A is described as

Ma = U Ml-

n^0

Multi-operation 0n of dimension n is described as empty operation if for all elements a1,...,an of A relation

en(au...,a,n) = 0

is true.

Multi-operation nn of dimension n is described as a complete poperation if for all elements a1,... ,an of A relation

nn(ai, ...,an) = A

is true.

Multi-operation en of dimension n is described as a projection multi-operation with respect to i-th argument if for all elements a1,... ,an of A relation

en(ai, ...,an) = {ai}

is true.

Let us note that multi-operation ern can be also considered as operation on A. Superposition for f G MA and fi G M™, (i = 1,... ,n), described as

(f * fl ,...,fn),

is defined as follows

(f * fi,...,fn)(ai,...,am)= [J f (bi ,...,bn)

biEfi{ai,...,am)

for all a1:..., am of A.

If f,f1,...,fn are operations then we have definition of operation superposition. Every subset K C PA is described as a clone on A if it contains all projection operations, and it is closed with respect to superpositions.

Every subset R C MA is described as a multi-clone on A if it contains all empty, complete multi-operations, projection multi-operations, and it is closed with respect to superpositions.

1. Super-clones

Solvability with respect to i-th argument for an n-ary multi-operation f is such a multioperation if ) that for all ai,... ,an of A relation

(Vif )(ai, ...,an) = {a\ai G f (a1:. .., a—i, a, a,i+1,..., an )}

is satisfied.

Substitution of a multi-operation g into the place of i-th argument of a multi-operation f is such multi-operation (f g) that relation

(f *i g)(ai,. .. )= U f (ai,...,ai-i ,b,ai+m,...,an+m-i)

b€g(ai,...,ai+m-1)

is satisfied.

Identification of i and j arguments of a multi-operation f is such a multi-operation (ai,jf ) that for all ai,..., aj^i, aj+i,... ,an of A relation

i,j f )(a1, . . ., aj-1, aj+1, .. ., an) = f (a1, . . . , aj-1, ai, aj+1: .. ., an)

is satisfied.

Intersection of multi-operations f and g from is such a multi-operation (f n g) that for all ai , . . . , an of A relation

(f n g)(ai, ...,an) = f (ai,.. ., an) n g(ai,. .., an)

is satisfied.

Let us note that by analogy with clones multi-clone can be defined as any subset R C M A [4]. A multi-clone contains all empty, complete multi-operations, projection multi-operations, and it is closed with respect to substitutions and identifications.

Lemma 1. For a set of multi-operations A that contains all empty, complete multi-operations, projection multi-operations the following conditions are equivalent:

1) A is closed with respect to superpositions and solvabilities;

2) A is closed with respect to substitutions, solvabilities and identifications;

3) A is closed with respect to substitutions, solvabilities and intersections.

Proof. Equivalence of 1) and 2) follows from representation of superposition in terms of substitutions and identifications of arguments, and permutation of arguments i and j of a multioperation f is expressed as

(Vif )).

Equivalence of 2) and 3) follows from equality

fn = (((n1 *i (en n en)) n fn) j n0).

Equivalence of 3) and 1) follows from identity

(f n g) = ( fn * f,g), rgefn = (el * e?, (p2el)).

□

A set is described as a super-clone if it satisfies one of the equivalent conditions of Lemma 1.

2. Semi-identity of superposition solvability

To prove the equality of super-clones and multi-clones one should transfer solvability operators through superposition. However, the possibility of such operation is still not proved. In the following lemma we give only the identity inclusion (semi-identity) and show that the identity is not satisfied.

Lemma 2. The following semi-identity is satisfied:

&(fn *gm, ...,gm) C H (Vig? *em, ■■■, e?-i, (Vj fn *nm, ..., em,, nm,..., nm), em+1, ..., e^ ). j=i " "

If f is unary multi-operation then the following identity is satisfied:

em

Vi(f * g?) = (Vig? * e?,..., e?L 1, (vf * e?), e?+i, ...,e??).

Proof. Let a G Pi(fn * g■ ■■, ■ ■ ■, bm). It follows from the definition of solvability

that

bi G (fn * gm,..., gm )(bi,..., bi-i, a, bi+i,bm).

Then it follows from the definition of superposition that there are x1,... ,xn such that

bi G fn(xi,...,xn),

xj G gji(bi,..., bi-i, a, bi+i,..., bm ),

for j = 1,... ,n.

Using solvabilities with respect to various arguments, we obtain

a G ¡Jigji(bi,..., bi-i ,xj, bi+i,..., bm), xj G pjfn(xi,.. .,xj-i,bi,xj+i, .. .,xn),

for j = 1,... ,n. Then

a G (pigm * em,...,em-i, (Pj fn * nm,...,nm, em ,nm,.. . ,nm),em+i, .. . ,em)(bi, ... ,bm),

for every j = 1,... ,n. Thus we have

g n (Pigm * em

j=i

jm <, xn , ^m

>ei-^ (Pj j *n ,

mm

ei+1,

..,em)(bi,...,bm).

Obviously, when n = 1 all reverse consequences are satisfied, and hence the identity holds. □ The following example shows that the reverse inclusion is not always true. Let us use vector representation of multi-operations [3]. Let f2 = (501042013), g\ = (465), gi = (736). Then

Pi(f2 * gi,gi) = (752), (Pigi * (Pif2 * e i,ni)) n (pigi * (p2f2*,n\ei)) = (756).

We obtain

P i (f * g\,gl ) C (p i g i * (p if2 * e i,ni )) n (p i g\ * (p2 f2*,n i,ei )).

3. Equality of multi-clone and super-clone

Below identities for the transfer of the solvability operator inside the term are found. This is appropriate for terms over intersection and substitution. In what follows brackets that are uniquely recovered are removed.

Lemma 3. The following identities are satisfied:

1) Pi(f n g) = pif n pig;

2) Pi(fn *j gmm = (Pifn *j gm) for i G {1, ...,j - 1,j + m,.., n + m - 1};

3) pi(fn *j gm) = an+m-i(pigm *i Pjfn) for i G {j, ...,j + m - 1}; where an+m-i is some transposition of arguments.

m m m

a

Proof. 1) Let for all a1,... ,an relation

a G Vi(f n g)(ai,. .., an) is satisfied. According to the definition of Vi, there is a relationship

ai G (f n g)(ai,. .., a-i, a, ai+i, ... ,ari)

Then we have ai G f (ai,..., ai-i, a, ai+i,..., an) and ai G g(ai,..., ai-i, a, ai+i,..., an), and also a G vif (ai,..., an) and a G vig(ai,..., an). Thus, we obtaine the following condition

a G Vif (ai, ...,an) n Vig(ai, .. ., an).

This condition is equivalent to the original condition, and equality 1) is proved.

2) Let for all ai,..., an+?-i relation

a G Vi(fn *j g?)(ai,. .., an+?-i)

is satisfied, where i G {1,... ,j — l,j + m,... ,n + m — 1}. According to the definition of vi, relation

ai G (fn *j g?)(ai,..ai-l, a ai+l,.an+?-i) is also satisfied. According to the definition of *j, there is an element a0 such that a0 G g?(aj,..., aj+?-i ) and ai G fn(ai,..., aj-i, ao, aj+?,..., am+?-i), where i G {1,.. .,j — 1,j + m,... ,n + m — 1}. These conditions are equivalent to the following conditions:

ao G g?(aj,..., aj+?-i) and a G Vif n(ai,..., aj-l, ao, aj+?,..., an+?-i).

Thus, we obtain

a G (Vifn *j g?)(ai,..., an+?-i).

*j

This condition is equivalent to the original condition. Equality 2) is proved.

3) Let for all ai,..., an+?-i relation

a G Vi(fn *j g?)(ai,..., an+?-i) is satisfied, where i G {j,... ,j + m — 1}. According to the definition of Vi, relation

ai G (fn *j g?)(ai,..ai-l, a ai+l,.an+?-i)

is also satisfied. According to the definition of *j, there is an element a0 such that a0 G

g?(aj,... ,a,..., aj+?-i) and i

ai G fn(ai, .. ., aj-i,ao, aj+?, .. ., ari+m-i).

Hence we have a G Vig?

(aj,. .., ao,. .., aj+?- i ) and i

ao G Vjfn(ai,.. ., aj-i,ai, aj+?,. .., ari+m-i).

Then

a G (Vig? *i Vj fn)(aj,. .., a-i, ai, .. ., aj-i, ai, aj+?, .. ., an+?-i,ai+i,. .., aj+?-i). Thus, we obtain

a G (Vig? *i Vj fn)(aj,. .., a-i, ai, .. ., aj-i, ai, aj+?, .. ., an+?-i,ai+i,. .., aj+?-i).

This condition is equivalent to the original condition for transposition of elements ai,..., an+?-i. Equality 3) is proved. □

Theorem. Let us assume that a set of multi-operations R contains multi-operation e2 and it is closed with respect to solvabilities. If multi-clone and super-clone are generated by R then they are equal.

Proof. In what follows we use standard concept of term over the set {*i,pi, n}. The notation &[fi, ■ ■ ■, fk] means that term Ф depends on fi, ■ ■ ■ ,fk.

Let us assume that an arbitrary multi-operation g is represented by term ■ ■ ■, fk] in a super-clone, where fs £ R, s = 1, ■ ■ ■ ,k. Using identities of Lemma 3, we transform term Ф into term ^[hi, ■ ■ ■ ,hr ] in which p can occur only for hj, where hj £ R, j = !,■■■, r. According to conditions of the theorem, pnhj £ R. Since (p2e2) £ R the intersection is expressed by a term because (f n g) = (fn * f, g), where fn = ( e2 * e2, (p2e2)). Thus, we obtain representation of g by the term over R without the use of the condition of closure with respect to solvability. □

The work of the second author was supported by the grant of the Buryat State University.

References

[1] D.Lau, Function Algebras on Finite Sets, Springer-Verlag, Berlin YeideWater Resourses Research, 2006.

[2] N.A.Peryazev, Clones, co-clones, hyperclones and superclones, Uchenie Zapiski Kazan. Univer. Ser. Fiz. Matem. Nauki, 151(2009), no. 2, 120-125 (in Russian).

[3] N.A.Peryazev, I.K.Sharankhaev, Galois theory for clones and superclones, Diskr. mat., 27(2015), no 4, 79-93 (in Russian).

[4] A.I.Maltsev, Iterative algebras and Post varieties, Algebra i Logika, 5(1966), no. 2, 5-24 (in Russian).

Об одном достаточном условии равенства мультиклона и суперклона

Николай А. Перязев

Санкт-Петербургский государственный электротехнический университет

Попова, 5, Санкт-Петербург, 197376

Россия

Иван К. Шаранхаев

Институт математики и информатики Бурятский государственный университет Смолина, 24а, Улан-Удэ, Респ. Бурятия, 670000

Россия

Рассматриваются мультиклоны и суперклоны, которые являются обобщениями таких стандартных объектов, как клоны. Суперклон получается из мультиклона добавлением условия замкнутости относительно 'разрешимости простейшего уравнения. В статье доказано условие, при котором мультиклон и суперклон совпадают.

Ключевые слова: мультиоперация, мультиклон, суперклон, суперпозиция, операция, подстановка.