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2Homomorphisms and Congruence Relations for Games with Preference Relations
Tatiana F. Savina
Saratov State Technical University,
Institut razvitiya biznesa i strategiy (Business and Strategy Development Institute)
Gorkogo st. 9, Saratov, 410028, Russia E-mail: suri-cat@yandex.ru
Abstract In this paper we consider games with preference relations. The main optimality concept for such games is concept of equilibrium. We introduce a notion of homomorphism for games with preference relations and study a problem concerning connections between equilibrium points of games which are in a homomorphic relation. The main result is finding covariantly and contravariantly complete families of homomorphisms.
Keywords: homomorphism, equilibrium points, Nash equilibrium, game with preference relations.
1. Introduction
In this paper we study games in which a valuation structure by preference relations is given.
We can consider a n-person game with preference relations as a system of the form
G = (Xi,..., Xn, A, pi,..., pn, F) (1)
where Xi is a set of strategies of player i (i = 1,... ,n), A is a set of outcomes,
Pi Q A2 is a preference relation of player i (i = 1,...,n) and realization function F is a mapping of set of situations X = X1 x ... x Xn in set of outcomes A.
The main optimality concept for games of this class are various modifications of Nash equilibrium. We introduce a concept of equilibrium as a generalization of Nash equilibrium for games of the form (1). We consider equilibrium and Nash equilibrium as optimal solutions for games with preference relations. The basic subject of research in our paper are homomorphisms of certain types. It is important that homomorphisms preserve optimal solutions of some types. The main results of the present work are theorems concerning connections between optimal solutions of games which are in a homomorphic relation.
2. Preliminaries
2.1. Basic concepts for preference structures
A preference structure on a set A can be given as a pair (A, P) where P is arbitrary
reflexive binary relation on A.
The condition (a1, a2) G p means that element a1 is less preference than a2. Given a preference relation p Q A2, we denote ps = p n p-1 its symmetric part and p* = p\ps its strict part.
We write
p
ai < a2 instead of (ai, a2) G p, ai 'Pp a2 instead of (ai, a2) G ps,
p
ai < a2 instead of (ai, a2) G p*.
pp
Remark 1. Conditions ai < a2 and a2 < ai are not compatible.
In this paper we consider some important types of preference structures: transitive, antisymmetric, linear, acyclic, ordinal.
Definition 1. A preference structure (A, p) is called
— transitive if for any ai, a2, a3 G A
(ai, a2) G p A (a2, a3) G p ^ (ai, a3) G p;
— antisymmetric if for any ai, a2 G A
(ai, a2) G p A (a2, ai) G p ^ ai = a2;
— linear if for any ai, a2 G A
(ai, a2) G p V (a2, ai) G p;
— acyclic if for any n = 2, 3,... and ai,..., an G A
(a1, a2) G p A . .. A (an—1, an) G p A (an, ai ) G p ai a2 . . . an ;
— ordinal if axioms of transitivity and antisymmetry hold.
Remark 2. An ordinal preference structure (A, p) is a transitive and acyclic one and the converse is true.
Thus, transitive preference structure and acyclic one are a natural generalization of ordinal preference structure.
Definition 2. Let (A, p) be a preference structure and e be an equivalence relation on A. Relation p is said to be acyclic under e if for any n = 2, 3,... the implication
ppppp see
ao < ai < a2 < ... < an < ao ^ ao = ai = ... = an
holds.
2.2. Homomorphisms of preference structures
Let (A, p) and (B, a) be two preference structures.
Definition 3. A mapping ^: A ^ B is called a homomorphism of the first structure into the second one if for any ai, a2 G A the condition
P a
ai < a2 ^ ^(ai) < ^(a2) (2)
A homomorphism ^: A ^ B is said to be a homomorphism ”onto” if ^ is a mapping of A onto B.
A homomorphism ^ is said to be strict if the following two conditions are satisfied:
P &
ai < a2 ^ ^(ai) < ^(a2), (3)
ai £ «2 ^ ^(ai) £ ^(«2). (4)
A homomorphism ^ is called regular if the following two conditions
a P
^(ai) < ^(a2) ^ ai < a2, (5)
^(ai) £ ^(0,2) ^ ^(ai) = ^2). (6)
hold.
Remark 3. For any homomorphism the condition (4) holds. Indeed, let ^ be a homomorphism from A into B and ai £ a2 holds. The condition ai £ a2 means
p p a a
that ai < a2 and a2 < ai. Hence, ^(ai) < ^(a2) and r^(a2) < ^(ai) hold, i.e.
^(ai) £ ^(a2).
Remark 4. Any strict homomorphism is a homomorphism but the converse is false.
Let (A, p) be a preference structure and e C A2 an equivalence relation.
Definition 4. A factor-structure for preference structure (A, p) is a pair (A/ e, p/ e) where we denote for any Ci, C2 G A/ e:
(Ci, C2) G p/ e (3ai G Ci, a2 G C2) (ai, a2) G p.
Lemma 1 (about homomorphisms of preference structures).
Let (A, p) be a preference structure, e be an equivalence relation on A.
Then
1. a canonical mapping ^: a ^ [a]£ is a homomorphism from preference structure (A, p) onto factor-structure (A/ e, p/ e);
2. a canonical mapping ^ is a strict homomorphism if and only if condition
P
ai < £ a2
ai £ ai
a2 P a2
a2 < ai
> ^ ai
a2
(7)
is satisfied;
3. a canonical mapping ^ is a regular homomorphism if and only if conditions
a1 e =e a2
a1 p < a2
a1 e a1
a2 e a2
a1 p < a2
a1 e a1
a2 e a2
a2 p < a1
V —— O'-] < an
(8)
> — a- = an.
(9)
hold.
Proof (of lemma).
p
1. Suppose a- < a2. Then according to definition of factor relation we have
p/e
[aije < [a2]e. Hence, ^ is a homomorphism. Since a canonical homomorphism is a homomorphism ”onto”, we obtain the proof of the part (1) of the Lemma.
2. Let a canonical homomorphism ^ be strict and the implication condition (17)
p
is satisfied. Suppose a1 < a2. Since a canonical homomorphism is strict by
p/e
condition of the Lemma then [a1]e < [a2]e holds. On the other hand from the
p p/e
condition a'2 < a- it follows that [a2]e < [a-]e. As [a1]e = [a-]e, [a2]e = [a2]e
then
p/e
[a-]e < [a2]e,
p/e
. [a2]e < [ai]e.
The last system of conditions cannot be true (because of remark 1). Hence, our
pp assumption is not true and since a1 < a2 we get a1 ~ a2.
Conversely, suppose that the condition (17) holds. We have to prove that a canonical homomorphism is strict. Indeed, take two elements a1, a2 for which
p p p/e
a1 < a2 takes place, hence a1 < a2. By the part (1) of this Lemma [a1]e < [a2]e
p/e
holds. We assume that [a2]e < [a1]e. Then there exist elements a1, a22 such
e e p
that a1 = a1, a'2 = a2, condition a'2 < a1 holds. In this case, all assumptions of condition (17) are satisfied and by (17) we have a1 £ a2, which is contradictory
p p/e p/e
to a1 < a2. Thus, [a2]e < [a1]e does not take place and we get [a1]e < [a2]e. So, the first condition of homomorphism (3) for canonical homomorphism is satisfied. By remark 3 ^ is a strict homomorphism.
3. Suffice to verify that for regular homomorphism ^ its kernel e^ satisfies (8) and (9). Suppose
' e
a1 = a2
p
a1 < a2
e^
a'1 = a1
/ e^
,a2 = a2
p pa
From a1 < a2 it follows that a1 < a2 then ^(a1) < ^(a2). Assume that ^(a1) £ ^(a2); by using (11) we get ^(a1) = ^(a2), i.e. a1 = a2 is in contradiction with our assumptions. Hence, ^(a1) < ^(a2) holds, i.e. ^(a[) < ^(a'2). p
By (10) we obtain a1 < a'2 which was to be proved.
Now suppose conditions of (9) hold. Since ^ is a homomorphism we have
^(a{) < ^(a2),
^(a2) < ^(a1).
Hence, ^(a1) £ ^(a2). By (11) we get ^(a1) = ^(a2), i.e. a1 = a2.
p/e p/e
Conversely, assume [a1]e < [a2]e. Then [a1]e < [a2]e that is there exist such
e e p p
elements a1, a'2 that a1 = a1, a'2 = a2 and a1 < a'2. The condition a'2 < a1
p/e p/e
does not hold otherwise [a2]e < [a1]e, i.e. [a2]e < [a1]e; it is contradiction
p e
(see remark 1). Hence a1 < a'2. The condition a1 = a'2 does not hold, hence the conditions
/ £
a1 = a2,
p
a1 < a<2,
— / a1 = a1,
— / a2 a<2.
hold. According to (8) we obtain a1 < a2.
Now verify (11). Suppose [a1]e £ [a2]e, i.e. there exist elements a1, a" = a1 and a'2, a'2 = a2 such that
1 a2 ,
aa < a'(.
Then according to (9) we get a1 = a'2, i.e. [a1]e = [a2]e, which was to be proved.
□
Lemma 2. Let {A, p) be a preference structure, e be an equivalence relation on A. Factor-structure of preferences {A/ e, p/ e) is transitive if and only if the inclusion
p o e o p C e o p o e
(10)
Proof (of lemma).
Suppose (a1, a3) G p o e o p. According to the definition of composition of binary relations, then there exist such elements a2, a'2 G A that (a1, a2) G p, (a2, a'2) G e, (a'2,a3) G p hold. Denote by C1 = [a1]e, C2 = [a2]e = [a2]e, C3 = [a3]e. According to the definition of factor-relation we have (C1,C2) G p/e, (C2,C3) G p/e; since the factor-relation is supposed to be transitive then (C1, C3) G p/ e. It means that for some a1 G C1, a'3 G C3, (a'1,a'3) G p is satisfied. As a1 = a1, a'3 = a3 we get (a1, a3) G e o p o e which was to be proved.
Conversely, let the inclusion (10) be held. Let us take three classes C1, C2, C3 G A/ e for which (C1, C2) G p/ e, (C2, C3) G p/ e. Then there exist the elements a1 G C1, a2 G C2, a2 G C2, a3 G C3 such that (a1, a2) G p, (a2, a3) G p. Since a2 = a2 we get (a1, a3) G p o e o p. Hence, according to (10), (a1, a3) G e o p o e. It means that there exist the elements ai, a3 G A such that (at, at) G e, (ai,a3) G /3, (a3, a3) G e. Then ([ai]e, [a3]e) G p/ e and as [a3]e = [a3]e = C3, [ai]e = [ai]e = Cx we get (C1, C3) G p/ e which was to be proved. □
Corollary 1. Let {A, p) be a transitive preference structure, e be an equivalence rela,tion on A. If at least one of the conditions p o e C e o p or e o p C p o e or e C p holds then factor-structure {A/ e, p/ e) is transitive.
Proof (of corollary).
1. Indeed, let for example the first inclusion p o e C e o p be satisfied. Then p o e o p C (p o e) o p C (e o p) o p = e o p2 C e o p C e o p o e. According to Lemma 2 factor-structure { A/ e, p/ e) is transitive.
2. Now let e C p be satisfied. Multiplying the inclusion e C p by p to the left
we have p o e C p o p = p2 C p C e o p. Multiplying initial inclusion e C p by p to
the right we obtain e o p C p o p = p2 C p o e. From the inclusions proved we have
p o e = e o p, i.e. relations p and e commute. From part (1) of the proof of this corollary it follows that { A/ e, p/ e) is transitive. □
Lemma 3. Let {A, p) be a preference structure, e be an equivalence relation on A. Factor-structure {A/ e, p/ e) is a,cyclic if and only if p U e is acyclic under e.
Proof (of lemma).
Remark 5. It is easy to verify that conditions
pUe pUe pUe pUe pUe pUe e e e
ao ^ a1 < a1 < ... < an < a'0 < ao ^ ao = a1 = ... = an (11)
and
pepepepe eee ao ^ a1 = a1 < a2 = a2 < ... = an < ao = ao ^ ao = a1 = ... = an (12)
are equivalent.
Let the condition of the implication (12) be held. Put Co = [ao]e = [ao]e, C1 =
[a1]e = [a1]e,...,Cn = [an]e = [an]e. According to the definition of factor-
relation we have (Co, C1) G p/ e, (C1, C2) G p/e,..., (Cn, Co) G p/e. Since factor-
relation is supposed to be acyclic then Co = C1 = ... = Cn. It means that £ £ £ ao = a1 = ... = an.
Conversely, let (12) be satisfied. Let us take classes Co, Ci,...,Cn G A/ e, for which (Co,Ci) G p/ e, (Ci,C2) G p/ e,..., (Cn,Co) G p/ e. Then there exist elements ao G Co, ai G Ci, ai G Ci, a2 G C2,..., an G Cn, a0 G Co such
that (ao, a'i) G p, (ai, a2) G p,..., (an, a0) G p; since ai = ai (i = 0, 1,...,n) we get
ao = ai = ... = an. It means that [ao]e = [ai]e = ... = [an]e. As Co = [ao]e, Ci =
[ai]e,. ..,Cn = [an]e we obtain Co = Ci = ... = Cn. This completes the proof of
Lemma 3. □
3. Games with preference relations
3.1. Homomorphisms of games with preference relations
Consider two games with preference relations for players {1,...,n}:
G = {Xi,...,Xn, A, pi,..., pn, F) and r = (Ui,...,Un, B, au...,an, &).
Definition 5. A (n + 1) system of mappings f = (y>i,..., n, 0) where for any
i = 1,...,n, i: Xi ^ Ui and 0: A ^ B is called a homomorphism from game G
into game r if the following two conditions are satisfied:
for any i = 1,...,n and ai, a2 G A
Pi Oi
ai < a,2 ^ 0(ai) < 0(a2), (13)
0 o F = & o (^i □ ...D^n). (14)
Remark 6. For any situation x = (xi,..., xn) of game G condition (14) means that 0(F(xi,.. .,xn)) = $(pi(xi), ^2(x2),.. .,^n(xn)).
A homomorphism f is said to be strict homomorphism if system of the conditions
Pi Oi
ai < an ^ 0(ai) < 0(an), (i = 1,...,n) (15)
ai ~ an ^ 0(ai) ~ 0(a2) (i =1,...,n) (16)
holds instead of condition (13).
A homomorphism f is said to be regular homomorphism if for any i = 1,...,n, mapping 0 is a regular homomorphism between the preference structures (A, pi) and (B,ai), that is the following two conditions
Oi Pi
0(ai) < 0(a2) ^ ai < a2, (17)
0(ai) ~ 0(an) ^ 0(ai) = 0(an). (18)
hold.
A homomorphism f is said to be homomorphism ”onto”, if each i (i = 1,...,n) is a mapping ”onto”; an isomorphic inclusion map, if each <pi (i = 1,...,n) is one-to-one function; an isomorphism, if for any i = 1,...,n, ^i is one-to-one function and mapping 0 is an isomorphism between (A, pi) and (B, ai), that is the following equivalence
Pi Oi
ai < a2 & 0(ai) < 0(a2) (19)
Definition 6. A (n + 1) system of equivalence relations e = (ei,... , e„, e) where £i c x2 (i = i,... ,n),£ c A2 is called congruence in game G if consistency condition for realization function holds, i.e.
x\ — x\
i £2 x2 = X2
— F(xi, . .., xn) — F(xi, . .., xn).
(20)
Congruence e in game G is said to be str-congruence if consistency condition for preference relations for any i = 1 ,...,n
ai < a2
/ — ai — ai
f — a2 a2
pi
a2 < af
> — ai
a2
(2i)
holds.
Congruence e in game G is said to be reg-congruence if the following two conditions for any i = 1 ,...,n
ai — a2 Pi
ai < a2 / £
ai £ ai
a2 Pi a2
ai < £ a2
ai £ ai
a2 Pi a2
a2 < ai
V —— af < a2
(22)
> — ai — a2.
(23)
hold.
Definition 7. Let f = (y>i,...,y>n, 0) be a homomorphism from game G into game r. A (n + 1) system of equivalence relations £f = (eVl,..., eVn, e^) where for any i = 1,...,n, eVi is kernel of ^i and e^ is kernel of 0, is called kernel of homomorphism f.
Theorem 1. Let G be a game with preference relations of the form (1) and e be a congruence in game G. Then we can define a factor-game G/ e with preference relations by
G/e= (Xi/ei,..., Xn/en, A/ e, pi/e,..., pn/e, Fe)
where realization function F— ([xi] ,..., [xn]— ) = [F(xi, ..., xn)]— .
1. Canonical homomorphism /g = (ifei, .. ., V’e) where for any i = 1,. .., n, ^—i: Xi ^ Xi/ ei and 0—: A ^ A/ e is a homomorphism from game G onto game G/e.
n
2. Canonical homomorphism f-g is strict if and only if congruence e is str-congru-ence.
3. Canonical homomorphism f-g is regular if and only if congruence e is reg-congru-ence.
The proof of this Theorem is based on Lemma 1.
Theorem 2. Let G and r be two games with preference relations and a (n +1) system of mappings f = (vi, ..., vn, 0) be a homomorphism of game G onto game r. Then
1. for a (n + 1)-tuple of equivalence relations £f = (eVl, . .., £Vrl, £,/,), where £f is kernel of homomorphism f, consistency condition (20) holds. Hence, we can construct factor-game G/ef;
2. there exists a (n + 1) system of mappings 9 = {9\,. .., 9n, 9) from game G/ef into game r which is an isomorphic inclusion map from G/ef into r.
Proof (of theorem).
1. Let the condition of the implication (20) be held. Since £/ is kernel of homomorphism f then for any i = 1 ,...,n
xi = xi & Vi(xi) = Vi(xi), a' — a & 0(a') = 0(a)
hold.
Let us prove that the equality 0(F(xi,...,x'n)) = 0(F(xi,...,xn)) is true. Since f is homomorphism, then 0(F(xi,. ..,x'n)) = ^(vi(x/i),..., vn(x'n)) and 0(F (xi,..., xn)) = @(vi(xi),..., Vn(xn)).
Thus, the equality ^(vi(x/i),..., vn(x'n)) = ^(vi(xi),..., Vn(xn)) is obvious. By using Theorem 1 we can construct factor-game G/ef and canonical homomorphism is a homomorphism from game G onto game G/ef.
2. We define isomorphic inclusion map 9 = (91,... ,6n, 9) from game G/ef into
game r by 9i([xi ] ) = Vi(xi) for any i = 1,...,n and 9([a]—^) = 0(a).
First, we prove that all mappings 9i,...,9n, 9 are one-to-one functions. For example, we verify that 9i is one-to-one function. We write
9i([xi] —n ) = 9i([xi]—n ) & Vi(xi) = Vi(xi) & xi =1 xi & [xi]—?! = [xi]—n.
Now we prove that 9 = (9\,..., 9n, 9) is a homomorphism from game G/ef into
game r. Suppose ([ai], [a2]G pi/e^ then there exist ai — ai,a'2 — a2 (i.e. 0(ai) = 0(ai), 0(a2) = 0(a2)) such that (ai,a2) G pi. Since f is a homomorphism, it follows that (0(a[),0(al2)) G ffi, that is (0(ai),0(a2)) G ai. By definition 9, we get (9([ai] ^ ),9([a2] ^ )) G ai. Hence, condition of homomorphism (13) for 9 holds.
Now we verify condition (14). We write
9(F— ([xi] — n1 , . . . , [xn] — )) 9([F (xi , ..., xn)] — ^ ) 0(F (xi ,..., xn)) .
Since f is a homomorphism then
0(F (xl, ...,xn))= $(Vi (xi ), ..., Vn (xn)) = ^(9i([xi] —V1 ),..., 9n([xn] )).
Thus, 0 = (01,..., On, 0) is an isomorphic inclusion map from factor-game G/ £ f into game r. This completes the proof of Theorem 2.
□
Theorem 3. Let G be a game with preference relations of the form (1) and e be a congruence in game G. A factor-game G/e is a game with transitive preference structure if and only if for any i = 1,...,n the condition
Pi ◦ e o pi C e o pi o e
holds.
The proof of this Theorem is based on Lemma 2.
Theorem 4. Let G be a game with transitive preference structure, e be a congru-
ence in game G. If for any i = 1,...,n at least one of the conditions pi o e C e o pi or e o pi (1 pi o e or £ C pi holds then a factor-game G/e is a game with transitive preference structure.
The proof of this Theorem is based on Corollary 1 of Lemma 2.
Theorem 5. Let G be a game with preference relations of the form (1) and e be
a congruence in game G. A factor-game G/ £ is a game with acyclic preference structure if and only if for any i = 1,...,n, pi U e is acyclic under e, i.e. the implication
PiU— PiU— PiU— PiU— — — —
ao < ai < ... < an < ao — ao — ai — ... — an
holds.
The proof of this Theorem is based on Lemma 3.
It is easy to see that the following results are true.
Theorem 6. A (n+ l)-tuple of equivalence relations £ = (ei, . .., en, e) in game G is kernel of some homomorphism from game G into a game with preference relations if and only if ~£ is a congruence in game G.
Theorem 7. A (n+ l)-tuple of equivalence relations £ = (ei, . .., e„, e) in game G is kernel of some strict homomorphism from game G into a game with preference relations if and only ife is a str-congruence in game G.
Theorem 8. A (n+ l)-tuple of equivalence relations £ = (ei, . .., e„, e) in game G
is kernel of some regular homomorphism from game G into a game with preference
relations if and only ife is a reg-congruence in game G.
3.2. Equilibrium points in games with preference relations
Let G be a game with preference relations of the form (1). Any situation x G X can be given in the form x = (xi)i=ii...jn, where xi is the i-th component of x. For
xi G Xi, we denote by x || xi a situation whose i-th component is xi and other components are the same as in x.
Definition 8. A situation x G X is called an equilibrium point in game G if such
i = 1,...,n and xi G Xi for which the condition
F(x) < F(x || xi)
holds do not exist.
Nash equilibrium point is an equilibrium point x for which the outcomes F(x) and F(x || xi) are comparable under preference relation pi for any i = 1,...,n. In this case it satisfies
Pi
F(x || xi) < F(x).
Let K and K be two arbitrary classes of games with preference relations. Fix in these classes certain optimality concepts and let Opt G be the set of optimal solutions of any game G G K, Opt r the set of optimal solutions of any game r gK. If f is a homomorphism from G into r, then a correspondence between outcomes (and also between strategies and between situations) of these games is given; we denote this correspondence also by f.
Definition 9. A homomorphism f is said to be covariant, if f-image of any optimal solution in G is an optimal solution in r that is f (Opt G) C Opt r.
A homomorphism f is said to be contravariant, if f-preimage of any optimal solution in r is an optimal solution in G that is f-i(Optr) C OptG.
Now suppose that for each j G J a homomorphism fj of game G G K into some game rj gK is given.
Definition 10. A family of homomorphisms (fj)jeJ is said to be covariantly complete if for each x G Opt G there exists such index j G J that fj (x) G Opt rj.
A family of homomorphisms (fj )jeJ is said to be contravariantly complete if the condition fj (x) G Opt rj for all j G J implies x G Opt G.
Lemma 4. 1. A family of homomorphisms (fj )jeJ is a covariantly complete fam-
ily of contravariant homomorphisms if and only if
OptG = J f-i(Optrj). (24)
jeJ
2. A family of homomorphisms (fj)jej is a conravariantly complete family of covariant homomorphisms if and only if
OptG = p| f-i(Optrj). (25)
jeJ
Proof (of lemma).
We prove, for example, assertion 1. Since for each j G J, fj is a conravariant
homomorphism then by definition we get f-i(Opt rj) C Opt G. Hence, for arbitrary
family of conravariant homomorphisms
J f-i(Optrj) C OptG
jeJ
is satisfied. Since (fj)jeJ is covariantly complete family of homomorphisms then there exists such index j G J that fj (OptG) C Optrj, i.e. OptG C f-i(Opt T). Hence
OptG C J f-(Optrj).
jeJ
Thus,
OptG = J f-(Optrj).
jeJ
It is easy to verify that the converse is true. This completes the proof of Lemma 4.
□
Now consider the case when an optimality concept is the concept of equilibrium. It is easy to verify that the following result is true.
Theorem 9. 1. For equilibrium any strict homomorphism is a contravariant ho-
momorphism.
2. For equilibrium any regular homomorphism is a covariant homomorphism.
3. For Nash equilibrium any homomorphism ”onto” is a covariant homomorphism.
References
Birkhoff, G. (1967). Lattice theory. Amer. Math. Soc., Coll. Publ., Vol. 25.
Savina, T. F. (2009a). Mathematical Models for Games, Based on Preference Relations. Game Theory and Management. Collected abstracts of papers presented on the Third International Conference Game Theory and Management. SPb.: Graduate School of Management SPbU, pp. 227-228 Savina, T. F. (2009b). Covariant and Contrvariant Homomorphisms of Games with Preference Relations. Izvestya SGU, Vol. 9, No. 3, pp. 66-70 (in Russian)
