Minimal algebras of unary multioperations Текст научной статьи по специальности «Математика»

Journal of Siberian Federal University. Mathematics & Physics 2016, 9(2), 220-224

УДК 519.7

Minimal Algebras of Unary Multioperations

Nikolay A. Peryazev*

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

Russia

Yulia V. Peryazeva^

Gymnasium 24 of Saint Petersburg Srednii Avenue, 20, Saint Peterburg, 199053

Russia

Ivan K. Sharankhaev*

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

Russia

Received 10.01.2016, received in revised form 17.02.2016, accepted 24.03.2016 A matrix impression of algebras of unary multioperations of a finite rank and the list of the identities which are carried out in such algebras are gained. These results are used for the proof of the main result: descriptions of the minimal algebras of unary multioperations of a finite rank. As a result the list of all such minimal algebras for small ranks is received.

Keywords: multioperation, algebra, minimal algebra, matrix, operation, substitution. DOI: 10.17516/1997-1397-2016-9-2-220-224.

Introduction

Algebras of unary multioperations which are considered in this paper are finite algebras. Description of minimal algebras is important to study the structure of these algebras [1]. A description of all algebras of unary multioperations of rank 3 was obtained in [2]. The main result of this paper was announced in [3]. We note that algebras of unary multioperations are used for the study of the superclones and hence the clones [4].

Let B(A) be the set of all subsets of A. A mapping from A into B(A) is called unary multioperation on A. The set of all unary multioperations on A will be denoted by M\. Multioperation f on finite set A = {ao,..., ak-1} can be represented as mapping

f : {2°, 21,..., 2k-1} ^ {0,1,..., 2k - 1},

which is obtained from f by coding ai ^ 2®; 0 ^ 0; {ail,.. .,ais} ^ 211 + ■ ■ ■ + 2is.

And multioperation f is represented by vector (a°,..., ak-1), where f (a®) = a®, using the coding.

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

Let S C MA. Algebra F =< S; *, n,i,e,9,n > with operations of substitution (f * g), intersection (f n g), reversibility (if) and nullary operations e,9,n is called algebra of unary multioperations on A:

(f * g)(a) = {b| there exists c £ g(a) such that b £ f (c)}; (f n g)(a) = f (a) n g(a); (if )(a) = {bla £ f(b)};

e(a) = {a}; 9(a) = 0; n(a) = A.

The power of set A is called rank of algebra. Further we believe that rank is finite and equal k > 2.

We note some simple properties of operations of algebra of unary multioperations: f * (g * h) = (f * g) * h, f n (g n h) = (f n g) n h, f n g = g n f, i(if) = f, i(f n g) = if n ig, i(f * g)= ig * if, f * £ = £ * f = f, 9 * f = f * 9 = 9, f n n = f, f n 9 = 9, ne = e, = 9, ¡in = n. There is the following matrix representation of algebras of unary multioperations. Let B =< {0,1}; *, + > be two-element Boolean algebra. Boolean matrices are binary matrices on the elements which define the Boolean operations.

For unary multioperation f on A we define Boolean square matrix Mf = (a j) of order k as follows: aij = 1 if ai £ f (aj) else aij = 0.

Operations of algebra of unary of multioperations are represented by matrix operations in the following way:

Mf *g = Mf * Mg is matrix multiplication;

Mf ng = Mf o Mg is element-wise matrix multiplication;

M^f = Mj is transposition of matrix;

ME = E is diagonal matrix;

Mg = O is null matrix;

Mn = P is unit matrix.

For example, unary multioperation in vector form f = (3, 7,1) is represented by matrix

( 1 1 1 Mf = I 1 1 0

010 The main result

The smallest algebra which not equal trivial algebra consisting of only multioperations n, 9, e is called minimal algebra of unary multioperations. It is obvious that necessary and sufficient condition for minimality of algebra of unary multioperations is the generating of any its multioperation which not equal n, 9, e. The following theorem describes the multioperations generating minimal algebras of unary multioperations.

Theorem 1. Multioperation f on A which not equal n, 9, e generates minimal algebra of unary multioperations of rank k if and only if it satisfies one of the following conditions:

1) f n e = e, if = f, f2 = f;

2) f n e = e, ¡f = f, f2 = n;

3) f n e = if n f = e, f * if = if n f = n, f2 = f;

4) f n e = ¡if n f = e, f * if = if n f = n, f2 = n;

5) f n £ = 9, f = f, f2 = n;

6) f n £ = 9, ¡f = fp-1, fp = £, where p is simple divisor of k;

7) There exists not empty set B C A such that either f (a) = B for all a € A,

or f (b) = {b} for all b € B and f (a) = 0 for all a € A \ B, or f (b) = A for all b € B and f (a) = 0 for all a € A \ B, or f (b) = B for all b € B and f (a) = 0 for all a € A \ B.

Proof. The fact that the algebras generated by the multioperations f with these properties

will be minimal follows from the fact that

if conditions 1), 2), 5) are fulfilled then algebras consists of four elements n,9,£,f;

if conditions 3), 4) are fulfilled then consists of five elements n, 9, £, f, ¡f;

if condition 6) is fulfilled then consists of p + 2 elements n,9,£,f,f2,..., fp-1;

if condition 7) is fulfilled in case of one-element set A then consists of six elements

n, 9, £, (0,..., 0, 20,..., 0), (0,..., 0, 2k-1, 0,..., 0), (2®,..., 2®), i ® else consists of seven elements

n, 9, £, (2®1 + ■ ■ ■ + 2®s,..., 2®1 + ■ ■ ■ + 2®s), (0,..., 0, 2®1, 0,..., 0, 2®s, 0,..., 0),

®1 ®s

(0,..., 0, 2k-1, 0,..., 0, 2k-1, 0,..., 0), (0,..., 0, 2®1 + ■■■ + 2®s, 0,..., 0, 2®1 + ■■■ + 2®s, 0,..., 0)

®1 ®s ®1 ®s

(here we specify that for the last three components of the non-zero elements are in positions i1,...,is). In addition each multioperation other than n,9,£ generates all elements of its algebra.

We now show that any f generating a minimal algebra of unary multioperations will satisfy one of the seven conditions of the theorem. We consider the possible cases:

1. f n £ = £. It is clear that ( f2) C (f) and since f generates minimal algebra then it holds either ( f2} = (f) or ( f2} = {n, 9, £}. Since f n £ = £ then units of matrix Mf stored in matrix Mf 2. Hence in first case f2 = f, since else f € (f2), and in second case it is obvious that f2 = n.

1.1. If ¡f = f then first case corresponds condition 1) of the theorem, and second case — condition 2).

1.2. Let ¡f = f. By the properties of algebra operations multioperation g = f n ¡f has properties g n £ = £, g = ¡g. It is clear that (g) C (f) and since f generates minimal algebra then it holds either (g) = (f) or (g) = {n,9,£}. By g n £ = £, g = ng in first case we obtain (g) = {n, 9,£,g} = (f) that is impossible in view of f = g. From the second case implies f n nf = g = £. Similarly we obtain that multioperation h = f * nf has properties h n £ = £, h = ¡ih. Since (h) C (f) and f generates minimal algebra then it holds either (h) = (f) or (h) = {n,9,£}. As above, the first case is impossible, and in the second case we have f * ¡f = h = n. Equality ¡f * f = n is obtained analogously. In case f2 = f we obtain condition 3) of the theorem, and in case f2 = n - condition 4).

2. f n £ = 9. Consideration of the case is divided into two subcases.

2.1. if = f. In this case f2 n £ = £ since null rows are absent in matrix Mf 2 else algebra (f) contains a subalgebra satisfying condition 7) of the Theorem. Since (f2) C (f) and f generates minimal algebra then it holds either (f2) = (f) or (f2) = {n,9,£}. The first case is impossible since according to paragraph 1 would have received (f2) = {n,9,£,f2} or (f2) = {n, 9, £, f 2,if2}, but f = f2 and f = if2 because of f n£ = 9 and f2n£ = £, ¡f2n£ = £. In the second case we have f2 = n or f2 = £. The first version corresponds condition 5) of the

Theorem and the second version — condition 6) where p = 2.

2.2. if = f. By the properties of algebra operations multioperation g = f n if has properties g n e = 9, g = ig. Since (g) C (f) and f generates minimal algebra then it holds either (g) = (f) or (g) = {n,9,e}. In the first case since g n e = 9, g = ig we have (g) = {n,9,e,g} = (f), it is impossible because of f = g. In the second case since g n e = 9

k"2 — k

then g = 9. Hence f n if = 9. Thus units in matrix Mf no more —2—.

Multioperation h = f * if has properties h n e = e, h = ih. Since (h) C (f) and f generates minimal algebra then it holds either (h) = (f) or (h) = {n,9,e}. The first case is impossible because of f = h, and in the second case we have f * if = h = n or f * if = h = e.

But f * if = h = n is also impossible since because of f n e = 9 matrix Mf must have units k2 k

more —2—. We have f * if = e. Equality if * f = e is obtained analogously. From these equalities it follows that each row and each column of the matrix Mf has one unit, and it means that multioperation f is a permutation. Degrees of this permutation f,... ,fp respect to the operations *,i,e form a cyclic group which has no proper subgroups for simple p which is a divisor of k. Also it holds fp = e and if = fp-1. Since if = f then p > 3. This case corresponds condition 6) of the theorem for p ^ 3.

3. f n e = (0,..., 0,2h, 0,..., 0,2is, 0,..., 0). We consider the cases s = 1 and s > 2.

il is

3.1. f n e = (0,..., 0,2i, 0,..., 0). In this case algebra have minimal subalgebra which

i

contains three elements (0,..., 0,2i, 0,..., 0), (0,..., 0,2k-1,0,..., 0), (2i,..., 2i) in addition to

ii

n, 9, e, and it means that algebra is minimal only if f is equal one of these multioperations. It corresponds condition 7) of the theorem for one-element set B = {ai}.

3.2. f n e = (0,..., 0,2%1, 0,..., 0,2is, 0,..., 0). In this case algebra have minimal subalgebra

il is

which contains four elements (2il +-----+2is 2il +-----+2is), (0,..., 0, 2il, 0,..., 0, 2is, 0,..., 0),

il is

(0,..., 0, 2k-1,0,..., 0, 2k-1, 0,..., 0), (0,..., 0, 2il + ■■■ + 2is, 0,..., 0, 2il + ■■■ + 2is, 0,..., 0)

il is il is

in addition to n, 9, e, and it means that algebra is minimal only if f is equal one of these multioperations. It corresponds condition 7) of the theorem for set B = {ail,... ,ais}.

These arguments concludes the proof of the theorem. □

Using this theorem one can find all minimal algebras for small ranks. We will do it for rank k = 2,3,4. Also we will indicate type of multioperation which generating a minimal algebra of unary multioperations according to the number of properties in the theorem.

Minimal algebras of unary multioperations of rank 2 (total 4)

Minimal algebras of unary multioperations of rank 3 (total 18)

Type 1 does not exist Type 1 (1,6,6), (5,2,5), (3,3,4)

Type 2 does not exist Type 2 (7,3,5), (3,7,6), (5,6,7)

Type 3 (1,3) Type 3 (1,3,7), (7,2,6), (5,7,4)

Type 4 does not exist Type 4 (3,6,5)

Type 5 does not exist Type 5 (6,5,3)

Type 6 (2,1) Type 6 (2,4,1)

Type 7 (1,1), (2,2) Type 7 (1,1,1), (2,2,2), (4,4,4), (3,3,3),

(5,5,5), (6,6,6)

Minimal algebras of unary multioperations of rank 4 (total 86) Type 1: (1,14,14,14), (13,2,13,13), (11,11,4,11), (7,7,7,8), (1,2,12,12), (1,10,4,10), (1,6,6,8), (9,2,4,9), (5,2,5,8), (3,3,4,8), (3,3,12,12), (5,10,5,10), (9,6,6,9).

Type 2: (11,7,14,13), (13,14,7,11), (7,11,13,14), (15,3,5,9), (15,7,7,9), (15,3,13,13), (15,11,5,11), (3,15,6,10), (3,15,14,14), (11,15,6,11), (7,15,7,10), (5,6,15,12), (5,14,15,14), (6,6,15,13),

(7.7.15.12),(9,10,12,15), (9,14,14,15), (13,10,13,15), (11,11,12,15), (15,15,7,11),

(15.7.15.13), (15,11,13,15), (7,15,15,14), (11,15,14,15), (13,14,15,15).

Type 3: (1,3,7,15), (3,2,7,15), (5,7,4,15), (1,7,5,15), (7,2,6,15), (7,6,4,15), (1,3,5,15), (1,7, 7,15), (7,2,7,15), (3,2,6,15), (7,7,4,15), (5,6,4,15), (15,2,6,10), (15,2,6,14), (15,2,14,10),

(15.2.14.14), (15,6,4,12), (15,6,4,14), (15,14,4,12), (15,14,4,14), (5,15,4,12),

(5.15.4.13), (13,15,4,12), (13,15,4,13). Type 4: does not exist.

Type 5: (11,13,11,7), (6,13,11,6), (10,13,10,7), (12,12,11,7), (14,13,3,3), (14,5,11,5), (14,9,9,7). Type 6: (2,1,8,4), (4,8,1,2), (8,4,2,1).

Type 7: (1,1,1,1), (2,2,2,2), (4,4,4,4), (8,8,8,8), (3,3,3,3), (5,5,5,5), (6,6,6,6), (7,7,7,7), (9,9,9,9), (10,10,10,10), (11,11,11,11), (12,12,12,12), (13,13,13,13), (14,14,14,14).

References

[1] D.Hobby, R.McKenzie, The structure of finite algebras, Contemporary Mathematics, 76(1988).

[2] A.S.Kazimirov, N.A.Peryazev, Algebras of unary multioperations, International Conference Maltsev meeting, Novosibirsk, 2013, 156 (in Russian).

[3] N.A.Peryazev, Minimal algebras of unary multioperations, International Conference Maltsev meeting, Novosibirsk, 2015, 193 (in Russian).

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

Минимальные алгебры унарных мультиопераций

Николай А. Перязев

Санкт-Петербургский государственный электротехнический университет

Проф. Попова, 5, Санкт-Петербург, 197376

Россия

Юлия В. Перязева

ГОУ гимназия 24 Средний пр., 20, Санкт-Петербург, 199053

Россия

Иван К. Шаранхаев

Институт математики и информатики Бурятский государственный университет Смолин, 24а, Улан-Удэ, 670000 Россия

Получено матричное представление алгебр унарных мультиопераций конечного 'ранга и список тождеств, выполняемых в таких алгебрах. Эти результаты используются для доказательства основного результата: описания минимальных алгебр унарных мультиопераций конечного ранга. Как следствие, получен список всех таких минимальных алгебр для небольших рангов.

Ключевые слова: мультиоперация, алгебра, минимальная алгебра, матрица, операция, подстановка.