On some maximal clone of partial ultrafunctions on a two-element set Текст научной статьи по специальности «Математика»

УДК 517.9

On Some Maximal Clone of Partial Ultrafunctions on a Two-element Set

Sergey A. Badmaev*

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

Russia

Received 08.07.2016, received in revised form 02.11.2016, accepted 20.01.2017 Multifunctions on a two-element set are considered in this paper. Functions from finite set to set of all subsets of this set are called multifunctions. Partial functions, hyperfunctions, ultrafunctions, partial hyperfunctions and partial ultrafunctions are arised depending on the type of multifunctions and superposition. In this work the problem of description of clones (sets of function closed with respect to the operation of superposition and containing all the projections) of partial ultrafunctions is considered. We got a description of one maximal clone of partial ultrafunctions on two-element set by the predicate approach.

Keywords: multifunctions, ultrafunctions, maximal clones, lattice. DOI: 10.17516/1997-1397-2017-10-2-140-145.

Introduction

In the theory of discrete functions the classical problem is description of lattice of clones. Full description of a lattice is obtained only for Boolean functions [1,2]. Because of difficulty of this problem lattice fragments are studied, for example, the minimum and maximum elements, different intervals. In particular, we note that the descriptions of all maximal clones are known for functions of ¿¡-valued logic, partial functions of ¿¡-valued, hyperfunctions and ultrafunctions on a two-element and partial hyperfunctions on a two-element set [3-8]. Some maximal clones of partial ultrafunctions found in [9,10]. In this paper we got a description of one maximal clone of partial ultrafunctions on a two-element set.

1. Basic concepts and definitions

Let E2 = {0,1} u F = {0, {0}, {1}, {0,1}}. We define the following sets of functions:

Pl,n = {f If : En ^ F}, P2 = (J Pln,

n

P2 ,n = {f If e Pin end If(5)| = 1 for every 5 e E.n}, P2 = UP2 n

n

Functions from P2 are called Boolean functions, and functions from Pi are called multifunctions on E2.

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

By definition [9,10] we believe that the superposition

f (fi (x1}... 7 xm

where f, f1,..., fn G P|, represents some function g(x1,..., xm), if for every ..., am) G E™

p| f (pi,... 7 ¡3n), if the intersection is not empty; a(ai a ) - t ^fii^'-^m)

gya.ii...7 am) — ^ y f ), otherwise.

I3iefi(ai,...,am)

On the tuples containing 0, the multifunction takes the value 0.

The multifunctions are considered with this superposition are called partial ultrafunctions. Below partial ultrafunctions are called simply functions. The tuples which contains element from E2 are called binary tuples.

Projection is called n-ary function: eln : (ai,... ,ai,..., an) ^ {ai}. For the set of functions B closure [B] is defined as follows:

1. B U{en}C [B].

2. If f,fi,...,fn G [B], then f (fi, ...,fn) G [B].

3. In the [B] other functions are not exist.

A set of functions is called closed set, if it is equal to its closure. If closure of the set of functions B coincides with the closed set M, then B is called full set in the M.

A set of function closed with respect to the operation of superposition and containing all the projections is called clone.

If for the clone K do not exist clone Ki such that K C Ki C P2n, then the clone K is called maximal clone.

Let Rs is s-ary predicate defined on the set F. The function f(xi,...,xn) preserves the predicate Rs, if for every tuples (aii,..., asi),..., (ain,..., asn) G Rs the tuple

(f (aii,... 7 ain),. .. 7 f (asi,. .., asn))

also belongs to the predicate.

Pol(Rs) denotes of the set of functions which preserves the predicate Rs. For simplicity we use the following code:

0 ^ *7 {0} ^ O7 {1} ^ I7 {O71} ^ -.

The tuple (t17 ... 7Tn) G is called rectification of the tuple (ji7... 7 jn) G Fn, if for those i, for whom Yi — —, follows that Yi — Ti, and for the other i we have Ti G E2.

The function of P2 n define its values at the binary tuples, the vector of values write in a row or column, and binary tuples assume to be given in the natural order.

0 1 — 1

If h(x7y) — (001—), f(x,y) — (10 — 1)7 g(x7y) — (—00*), then entry J

00 —0 — V1 *J

0

\*J

means that superposition h(f (x7y)7 g(x7y)) is equal to the function (10 — *).

2. Auxiliary statements

Here are some auxiliary results.

R4

, where (a, в, Y, S) are all sorts of columns in

Lemma 1. Let functions f,f1,...,fs preserve a predicate Rm defined on the set F, g(x1,..., xn) is superposition of the f (f1,..., fs) and binary tuples (a},..., a™),..., (an,..., a^) belong Rm. Then tuple (g(a},..., an),..., g(a"i,..., am)) belongs Rm.

Proof. This follo ws from the fact that for every binary tuple (P1,... ,Pn) is performed

g(P1, ...,pn) = f (f№, ...,Pn),--., fs(Pi,..., Pn)). Q

Consider the predicate

/0 0 0 0 1 1 1 1 - a\

0 0 1 1 1 1 0 0 - p

0 10 110 10 - 7

V0 1 1 0 1 0 0 1 - sj

which a,P,7,S e F are simultaneously satisfy two conditions:

• in every column (a, P, 7, S) among a, P, 7, S least two assume the value *;

• in every column (a, P, 7, S)f, if 0 or 1 are found among a, P, 7, S, then all of them are not equal to —.

Proofs of the following Lemmas 2 and 3 are identical with the proofs of the corresponding assertions of the work [8].

Lemma 2. Let a function f (x1,... ,xi-1,xi,xi+1,... ,xn) preserve the predicate Rm and the variable xi is dummy. Then the function g(x1,..., xi-1, xi+1,..., xn), derived from the f after removal of dummy variable xi, preserve the predicate Rm.

Lemma 3. Let Rm is m-ary predicate and for every tuple (Pi,... ,Pil,... ,Pi2,..., Pis,..., Pn) from the Rm such that Pil, Pi2,..., Pis are only equal to *, following condition are performed: if tuple (71, ...,7i1 ,...,7i2,... ,7is ,...,7n) e Rm, then tuple (Si,.. .,Sh ,...,Si2,... ,Sis,.. .,Sri), where Sj = * for j e {i},i2,... ,is} and Sj = 7j for other j, also belongs to the predicate Rm. Then if g(x},..., xi-1, xi, xi+1,..., xn) preserve the predicate, then f (x},..., xi-1, xi+1,..., xn), derived from the g after removal of dummy variable xi, also belongs to the predicate Rm.

Corollary 1. Pol(R4) are closed with respect to addition and removal of dummy variables.

The proof of the following lemma is given in the works [10, 11].

Lemma 4. The following sets of functions coincide with Pi n:

1)[{(1*), (1-)}]; 2)[{(*0), (-0)}]; 3)[{(0-), (-1), (0*)}]; 4)[{(0-), (-1), (*0)}].

3. The main result

Theorem 3.1. The class Pol(R4) is a maximal clone of partial ultrafunctions.

Proof. Let us first show that Pol(R4) is a clone. By the Corollary 1, and due to the fact that the projections preserve the predicate R4 remains to prove that R4 is closed with respect to the operation of superposition. Proof by contradiction. Let

h(xi, ...,xn) = f (gi (xi,. . .,xn ),g2 (xi, .. .,xn),. .., gm(xi,. . .,xn)), where f,g1,...,gm are arbitrary functions of the class Pol(R4).

Suppose there are tuples ai — (a^ ... 7 ain), where i G {17 27 ^7 4}, such that (aj a j aj aj) G

/0\ 0 0 1

R4 for every j, but h

/<A

0 1

\0j

/0\ 1 0 0

1 0 0 0

fal\

Œ2 â3 W7 (0\ 1 1 1

G R4, i.e. h

1 0 1 1

1 1 0 1

/&1\

a2 a3

\à4J

(1\ 1 1 0

coincides with one of the following columns:

m

0i

VJ

i^2\

V2 02

V2J

, where among ni,nionly one

value *, and among n2,n2, 02,X2 least one value —, and also has values 0 or 1.

Since swapping rows in the predicate R4 does not change it, it is sufficient to consider the cases:

/0\ C0\ cv\

0 0 1

1 1 1

n 0

W

f a.s\ fa\

where either ^7^7^ G {071}, or n — n — 6 — —, and case hi;) — ( j, where ag|011} h S7I G {17 27 37 4}.

We note that the tuples ai — (ai17... 7 ain) do not contain *, otherwise tuple h

fal\

,2

be-

a'

\&4J

longs R4. Also we note that for i G {17 27 37 4}, if h(ai) — a G {071}, then among rectifications a"1 are not such in which h is equal to —. Indeed, suppose that exists rectification S of tuple a1 such that h(S) — —, then exists rectification T of the same tuple such, that h(T) — a G {071}. Then

(T\

T T

h

\aj

G R4, which contradicts Lemma 1, since (ôk,rk, ôk,rk) G R4 for every k G {1,... ,n}. We consider the above four options. Everywhere we obtain a contradiction to Lemma 1. 1. h r3 = 1 . For the first three rows we choose rectifications S1,S2,S3 such that in

0

a2 0

a3 0

\a4j 1

which h is equal to 0, and for the fourth row we choose rectification T4 such that (S^ S\ 7 S3^ Sj)* G

R4 for every k G {1,... ,n}. Then h

(al\

a2 a3

\a4J

0 0 0 1

0 0 0

W

2. h

f&1\ 0

a2 1

a3 1

WJ

. For the second, third and fourth rows we choose rectifications t)2 7 T37 T4

1

such that in which h is equal to 1, and for the first row we choose rectification or such that

(T1\ ( /0\ (*\

(Si,S2,S3,54) G R4 for every k &{1,...,n}. Then h

a:

\a4J

G

1 1 1

1 1 1

t

2

3. h

a2 53 \<54

( ^

V |, where either j,n,6 £ {0,1}, or j = n = 6 = —. For the first three rows 6

51 52 S3

we choose rectifications J1, 5 2, 53 such that h(5l) = pl = *, where i £ {1, 2,3}, and for the fourth row we choose rectification 54 such that (5\,5\,5k,5\)t £ R4 for every k £ {1,...,n}. Then

/oA

53

WJ

4. h

(p1

vl

where a g {0,1} h s,l £ {1,2,3,4}. If there is the rectification 5 of tuple

Sl in which value of h is equae to —, then there exists the rectification 5 of tuple 5s such that

h(5) = a. Then h

r*\ -

5 | =

5 a

W \a

£ R4 h (Sk, 5k,rk,rk)t £ R4 for every k £ {1,..., n}.

If this rectification does not exist, then there are two rectifications 5l,S2 of tuele al such

/J1\ r /oe /o\

that h(o1) = 0 and h(°2) = 1. In case, when a = 0 we have h

52 51

\o2J

£

1 0 0

1 0

\*J

where

t1 is rectification of tuple 5s in which value of h is equal to 0, and 52 is rectification of tuple 5s such that (^,Sl,rl,Tk)t £ R4 for every k £ {1,...,n}. In case, when a = 1 we have

/ 51\

52 t1

VoV

0 1 1 1

0 1 1

w

>, where t1 is rectification of tuple 5s in which value of h is equal to 1,

and a2 is rectification of tuple as such that (5\ ,5\, tI,rl)t £ R4 for every k £ {1,... ,n}.

We now show that Pol(R4) is a maximal clone. It is sufficient to prove that [Pol(R4) U{f}] = P2, where function f does not preserve the predicate R4, i.e. there are tuples a1 = (a\,..., aln),

/5^

where i £ {1, 2, 3, 4}, such that (a1, a2, a3, a4)t £ R4 for every j, but f

£ R4.

It is easy to verify that the functions (00),(01),(11),(10),(0*),(1*),(*0),(*1),(—),(0110),(1001) preserve the predicate R4. Therefore, in view of Lemma 4 it is sufficient to have one of the functions (0-),(1-),(-0),(-1).

Since for every j £ {1,...,n} column ( aj, a2, a3, a4)t coincides with one of the functions

{(0000), (0011), (0101), (0110), (1111), (1100), (1010), (1001), (----)}, then applying operation

of superposition to f and to these functions we obtain or f1(x,y) = (0001), or f2(x,y) = (0111), or f3(x, y) = n, 0, *), or immediately one of the functions (0-),(1-),(-0),(-1).

Via binary functions f1, f2 h f3 we get one of the required functions (0-),(-0), (-1):

0/0 -\ r 0 \ 0 r0 -\ - 0 r0 A M 0 r0 0\ r0\ 0 r- 0\ r-\

011-

0

-1

011-

1

10

0 n 00 0

/

W 1

00 10 \1 *

/

0 1

\*J

-0 -1 -1

0 0

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

2

2

h

P

h

£

References

[1] E.L.Post, Determination of All Closed Systems of Truth Tables, Bull. Amer. Math. Soc., 26(1920), 427.

[2] E.L.Post, Introduction to a General Theory of Elementary Proposition, Amer. J. Math., 43(1921), no. 4, 163-185.

[3] I.G.Rosenberg, Uber die Verschiedenheit Maximaler Klassen in Pk, Rev. Roumaine Math. Pures Appl., 14(1969), 431-438.

[4] V.V.Tarasov, Completeness Criterion for Partial Logic Functions, Problemy Kibernetiki, 30(1975), 319-325 (in Russian).

[5] Lo Czu Kai, Maximal closed classes in the set of partial functions on multi valued logic, Kiberneticheskiy Sbornik. Novaya seriya., 25(1988), 131-141.

[6] L.Haddad, I.G.Rosenberg, D.Schweigert, A Maximal Partial Clone and Slupecki-type Criterion, Acta Sci. Math., 54(1990), 89-98.

[7] V.I.Panteleyev, Completeness Criterion for Incompletely Defined Boolean Functions (in Russian), Vestnik Samar. Gos. Univ. Est.-Naucn. Ser., 68(2009), no. 2, 60-79.

[8] V.I.Panteleyev, Completeness Criterion for Incompletely Defined Partial Boolean Functions, Vestnik Novos. Gos. Univ. Ser.: Mat., mekhan., inf., 9(2009), no. 3, 95-114 (in Russian).

[9] V.I.Panteleyev, On Two Maximal Multiclones and Partial Ultraclones, Izvestiya Irk. Gos. Univ. Ser. Matematika., 5(2012), no. 4, 46-53 (in Russian).

[10] S.A.Badmaev, I.K.Sharankhaev, On Maximal Clones of Partial Ultrafunctions on a Two-Element Set, Izvestiya Irk. Gos. Univ. Ser. Matematika., 16(2016), 3-18 (in Russian).

[11] S.A.Badmaev, On Complete Sets of Partial Ultrafunctions on a Two-Element Set, Vestnik Buryat. Gos. Univ. Mat., Inf., 3(2015), 61-67 (in Russian).

Об одном максимальном клоне частичных ультрафункций на двухэлементном множестве

Сергей А. Бадмаев

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

Рассматриваются мультифункции на двухэлементном множестве. Под мультифункцией на конечном множестве понимается функция, определенная на данном множестве и принимающая в качестве значений его подмножества. В зависимости от вида мультифункций и соответствующей им суперпозиции возникают частичные функции, гиперфункции, ультрафункции, частичные гиперфункции и частичные ультрафункции. В заметке исследуется задача описания решетки клонов (множеств, замкнутых относительно суперпозиции и содержащих все функции-проекции) для частичных ультрафункций. С помощью предикатного подхода получено описание одного максимального клона частичных ультрафункций на двухэлементном множестве.

Ключевые слова: мультифункции, ультрафункции, максимальные клоны, решетка.