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ИРКУТСКИЙ ГОСУДАРСТВЕННЫЙ УНИВЕРСИТЕТ ПУТЕЙ СООБЩЕНИЯ
масштабирования параметров сигнала (для связей в виде рычага первого или второго рода) или масштабирования с одновременным изменением направления движения точек закрепления рычага с типовыми элементами ВЗС.
БИБЛИОГРАФИЯ
1. Вейц В. Л., Кочура А. Е. Динамика машинных агрегатов двигателями внутреннего сгорания. Л. : Машиностроение, 1976. 314 с.
2. Мямлин С. В. Моделирование динамики рельсовых экипажей. Днепропетровск: Новая идеология, 2002.240с.
3. Соколов М. М., Хусидов В. Д., Минкин Ю. Г. Динамическая нагруженность вагона. М. : Транспорт, 1981. 208 с._
4. Соколов М. М., Варавва В. И., Левит Г. М. Гасители колебаний подвижного состава : справ. М. : Транспорт, 1985. 216 с.
5. Математическое моделирование колебаний рельсовых транспортных средств / под ред. В. Ф. Ушкалова. - Киев : Наук. думка, 1989. 240 с.
6. Коган А. Я. Динамика пути и его взаимодействие с подвижным составом. М. : Транспорт, 1997. 326 с.
7. Елисеев С. В., Упырь Р. Ю. Мехатронные подходы в задачах вибрационной защиты машин и оборудования // Современные технологии. Системный анализ. Моделирование. 2008. Вып. 4 (20). С. 8-16.
Chunhua Li
УДК 519.6
ON FAZZY CONGRUENCES OF ABUNDANT SEMIGROUPS
Introduction. The concepts of fuzzy equivalent relations on a set were introduced by Murali [ 10 ] and Nemitz [ 11], respectively. Samhan [ 8 ] defined fuzzy congruence relations on semigroups. In 2001, Tan [15] studied fuzzy congruences on a regular semigroups. As a generalization of regular semigroups in the range of abundant semigroups, El-Qallali and Fountain [ 1 ] introduced abundant semigroups. After that, various classes of abundant semigroups are researched (see, [ 2, 12-14 ]). In this paper, we shall study fuzzy congruence classes on abundant semi-
7"*
groups which preserve the Green's*-relation L and R* with respect to the binary operation "*" (see, [7]). We call them fuzzy-* congruences. We shall proceed as follows: section 1 provides some known results. In section 2, we define fuzzy-* congruences on abundant semigroups and give some characterizations of fuzzy* congruences on such semigroups. The last section we consider fuzzy-* congruences on the satisfying regularity condition semigroups, and give homomor-phism theorems.
1. Preliminaries. Throughout this paper we shall use the notions and notations of [1,3-7]. Other undefined terms can be found in [ 9 ]. We provide some known results used repeatedly, as well as some notations.
Let X be a non-empty set. A fuzzy subset f of X is a function of X into the closed interval [0,1]. A fuzzy relation / on X is a map from X x X to [0,1]. For all x g X, f (x) can be thought of as the degree of membership ofx in f.
Definition 1.1[7]. Let / and v be fuzzy relations on a semigroup S . Then the product /i°v of / and vis defined by /u°v(a,b) = vxgS {/(a,x) av(x,b)} for all x, y g S, and /çv is defined by /(x, y) < v(x, y) for all x, y g S.
Definition 1.2[7]. A fuzzy relation / on a semigroup S is called a fuzzy equivalence relation on S if
(1) /(a, a) = 1 for all a g S ;
(2) /(a, b) =/(b, a) for all a, b g S ;
(3) /o / ç
Definition 1.3[7]. A fuzzy relation / on a semigroup S is called left compatible if /(xa, xb) > /(a,b), for all a, b, x g S. Dually, we can define right compatible.
Let / be a fuzzy equivalence relation on a semigroup S. For each a g S , we define a fuzzy sub-
УПРАВЛЕНИЕ В ТЕХНИЧЕСКИХ СИСТЕМАХ. МОДЕЛИРОВАНИЕ
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set jia of S as follows: jia (x) = j(a, x) for all x e S. A fuzzy equivalence relation ji on S which is (left, right) compatible is called a fuzzy (left,right) congruence on S. We denote by Cp the characteristic function of a binary relation p on S. Then, as is easily seen, p is a (left, right )congruence on S if and only if Cp is a fuzzy (left, right) congruence on S . Let ji be a fuzzy congruence on a semigroup S. Then S / u = {ja | a e S}. Define the binary operation "* "on S / j as follows: jia * jib = jab (Va, b e S), We can prove that S / ji is a semigroup with respect to the binary operation "* ".
Lemma 1.4[7]. Let ji be a fuzzy equivalence
relation on a semigroup S , and let a, b e S. Then jia = jb if and only if u(a, b) = 1.
Next, we recall some of the basic facts about the relations L* and R*.
Lemma 1.5[1]. Let S be a semigroup and a, b e S . Then the following statements are equivalent:
(1) aL'b (aR*b);
(2) for all x, y e S1 ax = ay(xa = ya) if and only if bx = by( xb = yb).
As an easy but useful consequence of Lemma 1.5, we have
Corollary 1.6[1]. Let S be a semigroup and
a, e = e2 e S .Then the following statements are equivalent:
(1) alLLe (aR'e );
(2) a = ae(a = ea) and for all
x, y e S1 ax = ay(xa = ya) implies bx = by(xb = yb).
Evidently, LL is a right congruence while R* is a left congruence. In an arbitrary semigroup, we have L ^ LL and R ^ R*. But for regular elements a, b, we get aL*b(aR*b) if and only if aLb(aRb). Following [6], H *is defined as the intersection of L*and R*, and D*is defined as the join of L*and R*, L*a (
T" T"* 7"* tt *
Ra , Ha ) denotes the L class ( R class, H class ) containing a. For convenience, we denote by a+ [ a*,a0 ] a typical idempotent R* related [LL related, H* related ] to a. And E(T) denotes the set of idem-potents of T, E (T) is the subsemigroup of T generated by E (T).
A semigroup S is called abundant if and only
if each L* class and each R* class contains at least one idempotent. An abundant semigroup S is called super-abundant if and only if each H* class contains an idempotent. Let S be a super-abundant semigroup.
Then, as is easily seen, D* = LLrR = RrLL (see,[6]). An abundant semigroup S is called satisfying regularity condition if E (S) is the regular subsemigroup of S . An abundant semigroup in which the idempo-tents commute is adequate. It is clear that adequate semigroups satisfy regularity condition .
Next, we recall from [1] that a semigroup ho-momorphism (f> : S ^ T is a good homomorphism, if
for all elements a, b of S , alLb implies a(LLb( and aR*b implies a(R*b(. We say that a congruence p on a semigroup S is a good congruence if the natural homomorphism from S onto S / p is good. For more details of good congruences, the reader is referred to [13]and [14].
2. Definition and Characterizations. The aim of this section is to introduce fuzzy-* congruences on abundant semigroups, and to give some properties of fuzzy-* congruences on such semigroups.
Definition 2.1. A fuzzy right congruence jl on
an abundant semigroup S is called a fuzzy-* right congruence on S if for any t e[0,1], a e S,
x, y e S\ such that j(ax, ay) > t ^ ji(a*x, a*y) > t; Dually, we can define a fuzzy-* left congruence. A fuzzy congruence ji on an abundant semigroup S is called a fuzzy-* congruence of S if it is both a fuzzy-* left and a fuzzy-* right congruence of S .
Proposition 2.2. Let S be an abundant semigroup. Then the following statements are true:
(1) CL* is a fuzzy-* right congruence of S and
CR* is a fuzzy-* left congruence of S ;
(2) If S is a super-abundant semigroup, then
CL* 0 CR* = CR* 0 CL-
Proof. (1). Obviously, CL, is a fuzzy right congruence of S . Let t e [0,1], a e S, x, y e S1, and C* (ax, ay) > t. If t = 0, then C*(ax, ay) = 0. If t > 0, then C*(ax,ay) = 1, that is, (ax,ay) e L*.
Again, since L* is a right congruence, we have a*xlLaxL*ayL*a*y, that is, (a*x, a*y) e LL, which implies that CL* (a* x, a*y) = 1.
ИРКУТСКИЙ ГОСУДАРСТВЕННЫЙ УНИВЕРСИТЕТ ПУТЕЙ СООБЩЕНИЯ
Thus CL, (a* x, a*y) > t, so that CL is a fuzzy-* right congruence of S . Similarly, we can prove that CR* is a fuzzy-* left congruence of S .
(2) Let a, b e S. If (a, b) e LL or (a, b) e R*, then C* o CR, (a, b) = CR. o CL*(a, b) = 1. It suffices to verify that
C* o CR, (a, b) = CR, o C* (a,b) for (a, b) g LL and
(a, b) g R*. Notice that
CL' ° CR* (a' b) = ^xeSMa,b}{C1- (a> x) A CR* (x, b)}
If (a, x) g L or (x, b) g R* for any x e S \{a, b}, then CL. ° CR, (a, b) = 0, and (a, b) g D*. Hence, (a, z) g R* or (z, b) g L for any z e S \{a, b}. That is, CR, °C*(a,b) = 0. On the other hand, if there is an element x of S \{a,b} such that (a, x) e LL and (x, b) e R*, then C* o CR. (a, b) = 1, and (a, b) e D*. Hence, (a, z) e R*, (z, b) e LL, for some z e S \{a, b}. That is, CR, ° C^. (a, b) = 1. It follows that CL ° CR, (a, b) = CR. ° CL (a, b) and so (2) holds.
Proposition 2.3. Let p be a congruence on an abundant semigroup S . Then p is a good congruence on S if and only if Cp is a fuzzy-* congruence on S .
Proof. Necessity. Let p be a good congruence on S . Then, as is easily seen, Cp is a fuzzy congruence on S . We prove now that Cp is fuzzy -* congruence. Suppose that Cp(ax,ay) > t, where t e [0,1], a e S, x,y e S\ If t = 0, then Cp(a*x,a*y) > 0.If t > 0, then Cp(ax,ay) = 1, that is, (ax, ay) e p. As p is good, it follows that
S * * \ yT S * * \ -t
(a x, ay) e p. That is, Cp(a x, ay) = 1 > t. Hence, Cp(ax,ay) > t ^ Cp(a*x,a*y) > t.
Simlarly, Cp (xa, ya) > t ^ Cp (xa+, ya+) > t. Therefore, Cp is a fuzzy-* congruence on S .
Sufficiency. Let Cp be a fuzzy-* congruence on S . Then p is a congruence on S . Next, we prove that p preserves the relations L* and R*. To see this, suppose that apxp = apyp, for a e S, x, y e S\ that is, (ax, ay) e p. Hence, Cp (ax, ay) = 1 > 1. Again, since Cp is a fuzzy-* congruence on S , we
have Cp(a*x,a*y) > 1. That is, Cp(a*x,a*y) = 1,
which implies that (a*x, a*y) e p. That is, a*pxp =
*
a pyp. This, together with the fact that ap = apa*p, implies that ap L*(S/p) a*p. Similarly,
ap R*(S / p) a+p . This completes the proof.
Theorem 2.4. Let /u be a fuzzy congruence on an abundant semigroup S and let U = {(a, b) e S x S | u(a, b) > t}, where t e [0,1].
Then /U is a good congruence on S if and only if / is a fuzzy-* congruence on S .
Proof. Necessity. Let u(ax, ay) > t, where
t e [0,1], a e S, x, y e S\ Then (ax, ay) e /U ■ If U is a good congruence on S , then (a*x, a*y) e /U, that is, u(a*x, a*y) > t. Similarly, u(xa, ya) > t ^ /(xa +, ya + ) > t. Therefore, u is a fuzzy-* congruence on S .
Sufficiency. Let u is a fuzzy-* congruence on S . Then /U is a congruence on S . Next, we prove that /U is good. Let a e S, x, y e S\ and a/x/ = aUyU, that is, (ax, ay) e U. Then U(ax, ay) > t. Since U is a fuzzy-* congruence, we have U(a*x, a*y) > t, which implies that
* * t * tt * tt
(ax, ay) e u . That is, a uxu = a U yU . This,
together with the fact that aU = aUa*U, implies that aU LL(S / U ) a*U. Similarly, aU R*(S / U ) a+U . The proof is completed.
Proposition 2.5. Let U be a fuzzy-* congruence on an abundant semigroup S , and for all ae S,
such that a*a+ = a+a*. Then the following statements are equivalent:
(1) (Va e S)u e E(S / u);
(2) (3e e E(S))Ma = Me.
Proof. (1) implies (2). Let a e S,
Ua e E(S / u). Then U = Ua * U = U^. By Lemma 1.4, u(a, a2) = 1 > 1. Since u is a fuzzy-* congruence on S, we have u(a*, a*a) > 1, that is, U(a*, a*a) = 1. Hence, by Lemma 1.4, jd, = Ua*a. Dually, u + = u +. Thus, ua = u 2 = u , + =u+, =
^ y ' a ' aa y ' a 'a ' aa a a ' aa a a
where a+a* e E(S).
= Ц + * Ц , = Ц + * Ц , = Ц
aa a a a a ,
*!
a a
(2) implies (1). This is obvious.
Corollary 2.6. Let u be a fuzzy-* congruence on an abundant semigroup S . Then the following statements are true:
УПРАВЛЕНИЕ В ТЕХНИЧЕСКИХ СИСТЕМАХ. МОДЕЛИРОВАНИЕ
(1) If S is a super-abundant semigroup, then S / ji is super-abundant with respect to the binary operation "*";
(2) If S is an adequate semigroup, then S / ji is adequate with respect to the binary operation "*".
Proof. (1). Let ji be a fuzzy-* congruence on
an abundant semigroup S , and let t e [0,1], a e S,
x, y e S1. Then, by Defintion 2.1, ju(ax, ay) > t ^ j(a x, ay) > t and ji(xa, ya) > t ^ ju(xa+, ya+) > t. Notice that S is a super-abundant semigroup, we can choose t = 1, and so jiaL*(S / ji)ja„, jaR*(S / ji)ja„.
Thus, jiaH*(S / ji)ja0. This means that S / jis superabundant with respect to the binary operation "*".
(2) Let j be a fuzzy-* congruence on an adequate semigroup S . Then, by Proposition 2.5, as is easily seen, E(S / j)is a semilattice. Therefore, S / j is adequate with respect to the binary operation "*".
3. Some special cases
Proposition 3.1. Let ji be a fuzzy-* congruence on an abundant semigroup S . Then j is a fuzzy-* congruence on E (S) .
Proof. Let ji be a fuzzy-* congruence on S . Then, as is easily seen, j is a fuzzy congruence on E (S) . Next , we prove that j is a fuzzy -*congruence on E (S). Let a e E(S) E S, x,y e E(S)' e S1, such that j(ax, ay) > t, where t e[0,1]. Then ju(a*x, a*y) > t. Dually, j(xa, ya) > t ^
Proof. Let ^a e E(S/V). Then there are
/// eE(S///, such that Va = / */
a ' where n e N • Hence, by Proposition 2.5,
we have A ' where ' = 1>2>- 'n' e' e_f_( S >' and thus /ia = /.*/. * ■■ • */ = . .. en e E(S) //.
Conversely, if / e E(S) / then /a = .e ' where ei e E (S ), i = 1,2,--- , n, and thus,
-и = Meier. .e„ = Me, * MBi *• • • *№ e„g E(S / M). Therefore, E(S / и) = E(S) / M .
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Theorem 3.3. Let j be a fuzzy-* congruence on the satisfying regularity condition semigroups S and for all a e S, a*a+ = a+a*. ThenS / ji satisfies regularity condition with respect to the binary operation "*".
Proof. Let / e E(S / /). Then there exist /a1 , /a," . /„ e E(S / /X sUCh that /a =
/ */ *. . . */an, where n e N +. By Proposition 2.5, we have /la = /le, and thus / =
i i
U * и * •• • * и = и ,
where
e e E (S ), i = 1,2,. . . , n. Again, since S satisfies regularity condition, we get that e1e2... en is regular.
Hence, there exists x e E(S), such that eiV. . en = (eie2-. . en) x(eiV. . enX and thus
U a =U
a r ee2 • • en
Meer .e n )x(e,e2...en ) Ua * Ux * U
On the other hand, by Theorem 3.2, we have j e E(S) / ji = E(S / j), and so E(S / j) is a regular subsemigroup of S / j .
Corollary 3.4. Let j be a fuzzy-* congruence on an abundant semigroup S , and for all a e S, a*a + = a+a*. Then there is a homomorphism from E(S) to E(S / j).
Proof. Let ji be a fuzzy-* congruence on S . Then, by Theorem 3.1, ji is a fuzzy-* congruence on
ji(xa+,ya+) > t, for a e e(s), x,y e E(S) . The proof is completed.
Theorem 3.2. Let j be a fuzzy-* congruence on an abundant semigroup S and for all a e S,
a'a + = a+a'. Then E(S/ j) = E(S) / j.
E ( S ) . We define a mapping и : E(S) ^ E(S) / и as follows : и#(а) = иа, for all a g e ( S ) . It is clear that m# is a homomorphism. By Theorem 3.2, m# is
a homomorphism from E(S) to E(S / j). The proof is
completed.
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ИРКУТСКИЙ ГОСУДАРСТВЕННЫЙ УНИВЕРСИТЕТ ПУТЕЙ СООБЩЕНИЯ
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Baogen Xu
Y^K 519.1
ON REVERSE SIGNED DOMINATION NUMBERS OF GRAPHS
1. Introduction
We use Bondy and Murty [1] for terminology and notation not defined here and consider simple graphs only.
Let G=(V,E) be a graph, ifv e V(G), then NG
(v) denotes the open neighbourhood of v in G, and NG [v] = Ng (v) U{v}for the close neighbourhood, dG (v) = |Ng (v)| is the degree of v in G. For simplicity, sometimes, NG (v), NG [v] and dG (v) are denoted by N(v), N[v]and d(v), respectively. IfS c V(G), then G[S] denotes the subgraph of G induced by S. S(G) and A(G) denote the minimum and maximum degree
of G, respectively. G will denote the complement of G.
In recent years, some kinds of domination in graphs have been investigated [2~5]. T.W. Haynes, etc.[3] survey the major research accomplishments on domination theory.
Definition 1.1.[5] Let G = (V, E) be a graph, a signed dominating function (SDF) of G is a function
f: V(G) ^{-1,+1} satisfying £ f (v) > 1for all verveN [u ]
tices u e V(G), and the signed domination number of
G is defined as ys(G)= min{ £ f(v) If is an SDF
veV (G)
of G}.
In this paper, we initiate the study of a new graph parameter.
Definition 1.2. Let G = (V, E) be a graph, a function f : V ^ {-1,+1} is called a reverse signed dominating function (RSDF) of G if £ f (v) < 0
veN [u ]
holds for every vertex u e V . The reverse signed domination number of G is defined as
yrs (G ) = max { £ f (v) If is a RSDF of G}.
veV (G)
By the above definition, we have the following lemma.
Lemma 1.3. Let G be a graph of order n, then
(1) Yrs (G) = -n if and only if G = Kn ;
(2) For any two disjoint graphs GjandG2,
Yrs (G1 U G2) = Yrs (G1) + Yrs (G2);
(3) Yrs (G) - n(mod 2) .
Let G be a graph, if f be a RSDF of a graph G, and S c V(G), for convenience, we write
f (S) = £ f (v).
veS
In [5] we determined the smallest signed domination number for a graph G of order n.
