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Regular ordered semigroups in terms of fuzzy subsets
Information Sciences 176 (2006) 3675–3693 www.elsevier.com/locate/ins
Regular ordered semigroups in terms of fuzzy subsets Niovi Kehayopulu *, Michael Tsingelis Department of Mathematics, University of Athens, Panepistimiopolis, 157 84 Athens, Greece Received 12 July 2005; received in revised form 28 December 2005; accepted 1 February 2006
Abstract Given a set S, a fuzzy subset of S (or a fuzzy set in S) is, by definition, an arbitrary mapping f : S ! [0, 1] where [0, 1] is the usual interval of real numbers. If the set S bears some structure, one may distinguish some fuzzy subsets of S in terms of that additional structure. This important concept of a fuzzy set was first introduced by Zadeh. Fuzzy groups have been first considered by Rosenfeld, fuzzy semigroups by Kuroki. A theory of fuzzy sets on ordered groupoids and ordered semigroups can be developed. Some results on ordered groupoids–semigroups have been already given by the same authors in [N. Kehayopulu, M. Tsingelis, Fuzzy sets in ordered groupoids, Semigroup Forum 65 (2002) 128–132; N. Kehayopulu, M. Tsingelis, The embedding of an ordered groupoid into a poe-groupoid in terms of fuzzy sets, Inform. Sci. 152 (2003) 231–236; N. Kehayopulu, M. Tsingelis, Fuzzy bi-ideals in ordered semigroups, Inform. Sci. 171 (2004) 13–28] where S has been endowed with the structure of an ordered semigroup and defined ‘‘fuzzy’’ analogous for several notions that have been proved to be useful in the theory of ordered semigroups. The characterization of regular rings in terms of right and left ideals is well known. The characterization of regular semigroups and regular ordered semigroups in terms of left and right ideals or in terms of left, right ideals
* Corresponding author. Address: Nikomidias 18, 161 22 Kesariani, Greece. Tel.: +30 2107249762. E-mail address: [email protected] (N. Kehayopulu).
0020-0255/$ - see front matter 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.ins.2006.02.004
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and quasi-ideals is well known as well. The characterization of regular le-semigroups (that is lattice ordered semigroups having a greatest element) in terms of right ideal elements and left ideal elements or right, left and quasi-ideal elements is also known. In the present paper we first give the main theorem which characterizes the regular ordered semigroups by means of fuzzy right and fuzzy left ideals. Then we characterize the regular ordered semigroups in terms of fuzzy right, fuzzy left ideals and fuzzy quasi-ideals. The paper serves as an example to show that one can pass from the theory of ordered semigroups to the theory of ‘‘fuzzy’’ ordered semigroups. Some of our results are true for ordered groupoids in general. 2006 Elsevier Inc. All rights reserved. 2000 Mathematics Subject Classification: 06F05; 06D72; 08A72 Keywords: Ordered groupoid; Ordered semigroup; Fuzzy subset; Regular ordered semigroup; Left (right) ideal; Fuzzy left (right) ideal; Quasi-ideal of an ordered groupoid; Fuzzy quasi-ideal of an ordered groupoid; Idempotent fuzzy set (subset)
1. Introduction and prerequisites The application of fuzzy technology in information precessing is already important and it will certainly increase in importance in the future. Our aim is to promote research and the development of fuzzy technology by studying the fuzzy ordered semigroups. The goal is to explain new methodological developments in fuzzy ordered semigroups which will also be of growing importance in the future. This paper can be a bridge passing from the theory of ordered semigroups to the theory of fuzzy ordered semigroups. According to this paper, fundamental notions of ordered semigroups, like the notion of von Neumann regular ordered semigroups, can be expressed in terms of fuzzy sets, and the corresponding properties, like the important conditions which characterize the regular ordered semigroups, remain true (with natural modifications) in case of ordered semigroups, as well. On the other hand, many results of ordered semigroups generalize results of semigroups (without order), by taking the trivial equality relation as the order of the semigroups. On the other hand, the application of fuzzy semigroups in information processing, like in fuzzy coding, fuzzy languages and fuzzy finite state machines, have been proved to be useful. Given a set S, a fuzzy subset of S (or a fuzzy set in S) is, by definition, an arbitrary mapping f : S ! [0, 1] where [0, 1] is the unit segment of the real line. If the set S bears some structure, one may distinguish some fuzzy subsets of S in terms of that additional structure. In fact, a fuzzy set is a class of objects with a continuum of grades of membership. Such a set is characterized by a membership (characteristic function) which assigns to each object a grade of membership ranging between zero and one [41]. This important concept of the fuzzy set was first introduced by Zadeh in [41]. Since then, many papers on fuzzy sets
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appeared showing the importance of the concept and its applications to logic, set theory, group theory, groupoids, real analysis, measure theory, topology, etc. Many notions of mathematics are extended to such sets, and various properties of these notions in the context of fuzzy sets are established. The concept of a fuzzy set introduced by Zadeh, was applied in [4] to generalize some of the basic concepts of general topology. Rosenfeld [40] was the first who considered the case when S is a groupoid. He gave the definition of a fuzzy subgroupoid and the fuzzy left (right, two-sided) ideal of S and justified these definitions by showing that a (conventional) subset A of a groupoid S is a (conventional) subgroupoid or a left (right, two-sided) ideal of S if the characteristic function fA : S ! ½0; 1jx ! fA ðxÞ :¼
1
if x 2 A
0
if x 62 A
is, respectively, a fuzzy subgroupoid or a fuzzy left (right, two-sided) ideal of S. Kuroki has been first studied the fuzzy sets on semigroups [26–31]. See also Liu’s paper [33] where ‘‘fuzzy’’ analogous of several further important notions, e.g. those of bi-ideals or interior ideals have been defined and justified in a similar fashion. A survey of selected papers by Zadeh have been published in [42], where his main field of interest is centered around linguistic modeling of complex systems, approximate reasoning, and the use of fuzzy logical and natural language tools in expert systems and other areas of man–machine interaction. Selected papers by Zadeh have been also recently published in [43], which contains his deeply influential contributions to the field of fuzzy sets and their applications to approximate reasoning and modeling. New tools are needed for solving more difficult social and biological problems. This type of mathematics will be capable of handling uncertainties, making decisions and modeling very large systems and networks which are complex, non-linear and distributive. For an important article by Zadeh towards a generalized theory of uncertainty we refer to [45]. In one of his recent papers Zadeh introduced a new idea to explore the relationship between probabilities and fuzzy sets (see [44]). One of the areas in which fuzzy sets have been applied most extensively is in modeling for managerial decision making. The fuzzy set theory can be applied to the area of human decision making, as well [46]. Interesting results on fuzzy set theory and its applications have been also published by Zimmermann in [47], where one can see the application of fuzzy technology in information processing which is already important. In [47] one can also find new methodological developments in dynamic fuzzy data analysis, which will also be of growing importance in the future. Granular computing refers to the representation of information in the form of aggregates, called granules. If granules are modeled as fuzzy sets, then fuzzy logics are used. This new computing methodology has been considered by Bargiela and Pedrycz in [1]. A presentation of updated trends in fuzzy set theory and its applications has been considered by Petrycz
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and Gomide in [38]. Results of research into the use of fuzzy sets for handling various forms of uncertainty in the optimal design and control of complex systems are presented by Ekel et al. in [5]. Both fuzzy and neural control technologies are firmly based upon the principles of classical control theory [37]. A systematic exposition of fuzzy semigroups by Mordeson et al. appeared in [35], where one can find theoretical results on fuzzy semigroups and their use in fuzzy coding, fuzzy finite state machines and fuzzy languages. Fuzziness has a natural place in the field of formal languages. The monograph by Mordeson and Malik [34] deals with the application of a fuzzy approach to the concepts of automata and formal languages. This is the first monograph systematically presenting developments in the field. Fuzzy sets on semigroups have been also considered by Kehayopulu et al. in [20] and by Kehayopulu and Tsingelis in [17–19,22]. A theory of fuzzy sets on ordered groupoids and ordered semigroups can be developed. We endow S with the structure of an ordered groupoid or semigroup and define ‘‘fuzzy’’ analogous for several notions that have been proved to be useful in the theory of ordered semigroups. Following the terminology given by Zadeh, if S is an ordered groupoid (resp. ordered semigroup), a fuzzy set in S (or a fuzzy subset of S) is any mapping of S into the real closed interval [0, 1]. Based on the terminology given by Zadeh, fuzzy sets in ordered groupoids have been first considered by Kehayopulu and Tsingelis in [21,23,24]. Moreover, each ordered groupoid can be embedded into an ordered groupoid having a greatest element in terms of fuzzy sets [23]. The concept of regular rings was introduced by von Neumann [36]. It is well known that the class of such rings plays an important role in the abstract algebra, in the theory of Banach algebras [39] and in the continuous geometry [36]. An interesting result is that the set of all transformations on a finite dimensional vector space over a field forms a regular ring. Kova´cs characterized the regular rings as rings satisfying the condition A \ B = AB for every right ideal P A and every left ideal B, where AB is the set of all finite sums of the form aibi; ai 2 A, bi 2 B [25]. Ise´ki studied the same for semigroups. He proved that a semigroup (S, Æ) is regular if and only if for every right ideal A and every left ideal B of S, A \ B = AB [6]. As a consequence, a commutative semigroup S is regular if and only if every ideal of S is idempotent [6]. Calais gave the following characterization of regular semigroups: A semigroup S is regular if and only if the right and the left ideals of S are idempotent and for every right ideal A and every left ideal B of S, the product AB is a quasi-ideal of S [3]. The concepts of l-semigroups (: lattice ordered semigroups) and right–left ideal elements in ordered groupoids are due to Birkhoff [2, p. 323, 328]. An l-semigroup is a semigroup S at the same time a lattice such that a(b _ c) = ab _ ac and (a _ b)c = ac _ bc for every a, b, c 2 S. An le-semigroup is an l-semigroup S having a greatest element ‘‘e’’ (i.e. e P a for each a 2 S). An le-semigroup S is called regular, if for each x 2 S, we have x 6 xex [7]. An le-semigroup S is
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regular if and only if a ^ b = ab, equivalently, a ^ b 6 ab for every right ideal element a and every left ideal element b of S [8]. A commutative le-semigroup S is regular if and only if the ideal elements of S are idempotent (cf. [8, Theorem 2]). An le-semigroup S is regular if and only if for every right ideal element x and every left ideal element y of S, we have x2 = x, y2 = y (which means that, the right and the left ideals of S are idempotent), and the product xy is a quasiideal element of S [9]. It was proved in [16] that an ordered semigroup S is regular if and only if for every right ideal A and every left ideal B of S we have A \ B (AB], equivalently, A \ B = (AB]. Moreover, an ordered semigroup (S, Æ, 6) is regular if and only if the right ideals and the left ideals of S are idempotent and for every right ideal A and every left ideal B of S, the set (AB] is a quasi-ideal of S. As a consequence, a duo and so a commutative ordered semigroup S is regular if and only if the ideals of S are idempotent [16]. The results on semigroups—without order—mentioned above can be also obtained applying either the corresponding results on le-semigroups based on ideal elements or the results on ordered semigroups based on ideals (cf. e.g. [9,16]). As a consequence, the condition ‘‘A \ B AB for each right ideal A and each left ideal B’’ also characterizes the regular semigroups [16]. For further information we refer to [9,16]. In this paper we first give the main theorem which characterizes the regular ordered semigroups in terms of fuzzy right and fuzzy left ideals. We prove that an ordered semigroup S is regular if and only if for every fuzzy right ideal f and every fuzzy left ideal g of S we have f ^ g f g, equivalently f ^ g = f g, where f g is the multiplication defined for fuzzy subsets. As a result, a duo, and so a commutative ordered semigroup is regular if and only if the fuzzy ideals of S are idempotent. Then, we introduce the concept of a fuzzy quasi-ideal of ordered groupoids and the concept of idempotent fuzzy set in ordered groupoids, and we characterize the regular ordered semigroups in terms of fuzzy right ideals, fuzzy left ideals, and fuzzy quasi-ideals. We prove that an ordered semigroup S is regular if and only if the fuzzy right ideals and the fuzzy left ideals of S are idempotent and for each fuzzy right ideal f and each fuzzy left ideal g of S, the product f g is a fuzzy quasi-ideal of S. Thus, in addition with the characterizations of regular ordered semigroups we already have, we obtain equivalent characterizations of regular ordered semigroups using the fuzzy sets. The characterizations we obtain correspond to the characterizations of regular semigroups and regular ordered semigroups due to Kova´cs, Ise´ki, Calais and Kehayopulu. It might be noted that the results on fuzzy groupoids or fuzzy semigroups can be obtained as an application of the corresponding results of fuzzy ordered groupoids or fuzzy ordered semigroups, respectively. Let (S, Æ, 6) be an ordered groupoid. By a fuzzy subset of S we mean a mapping f : S ! [0, 1]. For a 2 S, define Aa :¼ fðy; zÞ 2 S Sja 6 yzg.
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For two fuzzy subsets f and g of S, define 8 W minff ðyÞ; gðzÞg < f g : S ! ½0; 1ja ! ðy;zÞ2Aa : 0
if Aa 6¼ ;; if Aa ¼ ;.
We denote by F(S) the set of all fuzzy subsets of S. One can easily see that the multiplication ‘‘’’ on F(S) is well defined. We define an order relation ‘‘’’ on F(S) as follows: f g if and only if f ðxÞ 6 gðxÞ
8x 2 S.
Let (S, Æ, 6) be an ordered groupoid and {fiji 2 I} a family of fuzzy subsets of S. When we refer W to a ‘‘family’’ V we always assume that it is non-empty. Define the fuzzy subsets i2I fi and i2I fi of S as follows: ! _ _ fi : S ! ½0; 1ja ! fi ðaÞ :¼ supffi ðaÞg and i2I
^
i2I
fi : S ! ½0; 1ja !
i2I
^
i2I
!
fi ðaÞ :¼ inf ffi ðaÞg. i2I
i2I
If I is a finite set, say I = {1, 2, . . . , n}, then clearly n _ i¼1 n ^
fi ðaÞ ¼ maxff1 ðaÞ; f2 ðaÞ; . . . ; fn ðaÞg and fi ðaÞ ¼ minff1 ðaÞ; f2 ðaÞ; . . . ; fn ðaÞg.
i¼1
One can easily prove the following: _ ^ fi ¼ supffi g and fi ¼ inf ffi g. i2I
i2I
i2I
i2I
which means that the set of all fuzzy subset of S, endowed with the order ‘‘’’ defined above, is a complete lattice. For two fuzzy subsets f, g of S, we clearly have f ^ g = min{f, g} and f _ g = max{f, g} (and these are the two operations of the lattice (F(S), )).
2. Characterization of regular ordered semigroups in terms of fuzzy right ideals and fuzzy left ideals The characterization of regular rings, semigroups or ordered semigroups in terms of right ideals and left ideals is well known [25,6,16]. The characterization of regular le-semigroups in terms of right ideal elements and left ideal
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elements is known as well [8]. In this paragraph we give the analogous result by means of fuzzy sets. We prove that an ordered semigroup S is regular if and only if for each fuzzy right ideal f and each fuzzy left ideal g of S, we have f ^ g = f g, equivalently, f ^ g f g. Let (S, Æ, 6) be an ordered groupoid. For A S, we denote ðA :¼ ft 2 Sjt 6 h for some h 2 Ag. We have A (A]. If A B, then (A] (B], (A](B] (AB], ((A]] = (A] [10]. A non-empty subset A of S is called a right (resp. left) ideal of S if (1) AS A (resp. SA A) and (2) If a 2 A and S 3 b 6 a, then b 2 A. If A is both a right and a left ideal of S, then it is called an ideal of S [10]. We denote by R(a) (resp. L(a)) the right (resp. left) ideal of S generated by a (a 2 S). We have R(a) = (a [ aS] and L(a) = (a [ Sa] [10]. An ordered semigroup (S, Æ, 6) is called regular if for every a 2 S there exists x 2 S such that a 6 axa. Equivalent Definitions: (1) A (ASA] "A S. (2) a 2 (aSa] "a 2 S [12]. In particular, if the ordered semigroup S is a poe-semigroup, that is an ordered semigroup having a greatest element ‘‘e’’, then it can be easily seen that S is regular if and only if x 6 xex for all x 2 S. Proposition 1. If (S, Æ, 6) is an ordered groupoid and f1, f2, g1, g2 fuzzy subsets of S such that f1 g1 and f2 g2, then f1 f2 g1 g2. Proof. Let a 2 S. If Aa = ;, then (f1 f2)(a) :¼ 0 6 0 :¼ (g1 g2)(a). Let Aa 5 ;. Then _ ðf1 f2 ÞðaÞ :¼ minff1 ðyÞ; f2 ðzÞg; ðy;zÞ2Aa
ðg1 g2 ÞðaÞ :¼
_
minfg1 ðyÞ; g2 ðzÞg.
ðy;zÞ2Aa
We have minff1 ðyÞ; f2 ðzÞg 6 minfg1 ðyÞ; g2 ðzÞg
8ðy; zÞ 2 Aa .
ð1Þ
Indeed: Let (y,z) 2 Aa. Since y, z 2 S, f1 g1 and f2 g2, we have f1(y) 6 g1(y) and f2(z) 6 g2(z), then minff1 ðyÞ; f2 ðzÞg 6 minfg1 ðyÞ; g2 ðzÞg. By (1), we have _ _ minff1 ðyÞ; f2 ðzÞg 6 minfg1 ðyÞ; g2 ðzÞg; ðy;zÞ2Aa
ðy;zÞ2Aa
thus (f1 f2)(a) 6 (g1 g2)(a). Thus we have f1 f2 g1 g2. h
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By Proposition 1, the set F(S) of all fuzzy subsets of S endowed with the multiplication ‘‘’’ and the order ‘‘’’ is an ordered groupoid. Lemma 1. An ordered semigroup (S, Æ, 6) is regular if and only if for every a 2 S we have R(a) \ L(a) (R(a)L(a)]. Proof. ). Let S be regular. Then for each right ideal A and each subset B of S, we have A \ B (AB]. Indeed: A \ B ððA \ BÞSðA \ BÞ ððASÞB ðAB. Let now a 2 S. Since R(a) is a right ideal of S, we have R(a) \ L(a) (R(a)L(a)]. (. Let a 2 S. Then a 2 RðaÞ \ LðaÞ ðRðaÞLðaÞ ¼ ðða [ aSða [ Sa ððða [ aSÞða [ SaÞ ¼ ðða [ aSÞða [ SaÞ ¼ ða2 [ aSa [ aS 2 a ¼ ða2 [ aSa. Then a 6 a2 or a 6 axa for some x 2 S. Then S is regular.
h
If (S, Æ, 6) is an ordered groupoid and A S, the fuzzy subset fA of S is the characteristic function of A defined as follows: ( 1 if x 2 A; fA : S ! ½0; 1jx ! fA ðxÞ :¼ 0 if x 62 A. Let (S, Æ, 6) be an ordered groupoid. A fuzzy subset f of S is called a fuzzy right ideal of S if (1) f(xy) P f(x) for every x, y 2 S. (2) If x 6 y, then f(x) P f(y). A fuzzy subset f of S is called a fuzzy left ideal of S if (1) f(xy) P f(y) for every x, y 2 S. (2) If x 6 y, then f(x) P f(y) [21]. A fuzzy subset f of S is called a fuzzy ideal of S if it is both a fuzzy right and a fuzzy left ideal of S. One can easily see that a fuzzy subset f is a fuzzy ideal of S if and only if the following assertions hold: (1) f(xy) P max{f(x), f(y)} for every x, y 2 S. (2) If x 6 y, then f(x) P f(y) [21].
Lemma 2 (Cf. [21]). Let S be an ordered groupoid. A non-empty subset L of S is a left ideal of S if and only if the characteristic function fL is a fuzzy left ideal of S.
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In a similar way we prove the following: Lemma 3. Let S be an ordered groupoid. A non-empty subset R of S is a right ideal of S if and only if the characteristic function fR is a fuzzy right ideal of S. Proposition 2. Let (S, Æ, 6) be an ordered groupoid, f a fuzzy right ideal and g a fuzzy left ideal of S. Then f g f ^ g. Proof. Let a 2 S. Then (f g)(a) 6 (f ^ g)(a). In fact: If Aa = ;, then (f g)(a) :¼ 0. Since a 2 S and f ^ g is a fuzzy subset of S, we have (f ^ g)(a) P 0, thus (f g)(a) 6 (f ^ g)(a). Let Aa 5 ;. Then ðf gÞðaÞ :¼
_
minff ðyÞ; gðzÞg.
ðy;zÞ2Aa
We prove that minff ðyÞ; gðzÞg 6 ðf ^ gÞðaÞ
8ðy; zÞ 2 Aa .
Then we have _ minff ðyÞ; gðzÞg 6 ðf ^ gÞðaÞ; ðy;zÞ2Aa
and (f g)(a) 6 (f ^ g)(a). Let now (y, z) 2 Aa. Then min{f(y), g(z)} 6 (f ^ g)(a). Indeed: Since (y, z) 2 Aa, we have y, z 2 S and a 6 yz. Then, since f is a fuzzy right ideal of S, we have f(a) P f(yz) and f(yz) P f(y), so f(a) P f(y). Since g a fuzzy left ideal of S, we have g(a) P g(yz) and g(yz) P g(z), so g(a) P g(z). Then minff ðaÞ; gðaÞg P minff ðyÞ; gðzÞg. So we have ðf ^ gÞðaÞ :¼ minff ðaÞ; gðaÞg P minff ðyÞ; gðzÞg.
Proposition 3. Let S be a regular ordered semigroup. Then for every fuzzy right ideal f and every fuzzy subset g of S, we have f ^ g f g. Proof. Let f be a fuzzy right ideal, g a fuzzy subset of S and a 2 S. Then (f ^ g)(a) 6 (f g)(a). In fact: Since S is regular, there exists x 2 S such that a 6 (ax)a. Then (ax, a) 2 Aa. Since Aa 5 ;, we have _ ðf gÞðaÞ :¼ minff ðyÞ; gðzÞg. ðy;zÞ2Aa
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Besides, (f ^ g)(a) = min{f(a), g(a)}. Since f is a fuzzy right ideal of S, we have f(ax) P f(a). Then min{f(ax), g(a)} P min{f(a), g(a)}. Thus we have ðf ^ gÞðaÞ 6 minff ðaxÞ; gðaÞg. Since (ax, a) 2 Aa, we have minff ðaxÞ; gðaÞg 6
_
minff ðyÞ; gðzÞg.
ðy;zÞ2Aa
Hence we have ðf gÞðaÞ ¼
_
minff ðyÞ; gðzÞg P minff ðaxÞ; gðaÞg P ðf ^ gÞðaÞ.
ðy;zÞ2Aa
Therefore f ^ g f g. h In a similar way we prove the following: Proposition 4. If S is a regular ordered semigroup, then for every fuzzy subset f and every fuzzy left ideal g of S, we have f ^ g f g. Theorem 1. An ordered semigroup S is regular if and only if for every fuzzy right ideal f and every fuzzy left ideal g of S, we have f ^ g f g; equivalently; f ^ g ¼ f g: Proof. Let S be regular, f a fuzzy right ideal and g a fuzzy left ideal of S. By Proposition 3, f ^ g f g. By Proposition 2, f g f ^ g. Then f ^ g = f g. Suppose f ^ g f g for every fuzzy right ideal f and every fuzzy left ideal g of S. Then S is regular. In fact: By Lemma 1, it is enough to prove that RðaÞ \ LðaÞ ðRðaÞLðaÞ
8a 2 S.
Let a 2 S, b 2 R(a) \ L(a). Then b 2 (R(a)L(a)]. Indeed: Since R(a) is a right ideal of S, by Lemma 3, fR(a) is a fuzzy right ideal of S. By Lemma 2, fL(a) is a fuzzy left ideal of S. Then, by hypothesis, ðfRðaÞ ^ fLðaÞ ÞðbÞ 6 ðfRðaÞ fLðaÞ ÞðbÞ. Since (fR(a) ^ fL(a))(b) :¼ min{fR(a)(b), fL(a)(b)}, we have minffRðaÞ ðbÞ; fLðaÞ ðbÞg 6 ðfRðaÞ fLðaÞ ÞðbÞ. Since b 2 R(a) and b 2 L(a), we have fR(a)(b) :¼ 1, fL(a)(b) :¼ 1, then we have min{fR(a)(b), fL(a)(b)} = 1, and 1 6 ðfRðaÞ fLðaÞ ÞðbÞ.
ð2Þ
If Ab = ;, then (fR(a) fL(a))(b) :¼ 0, which is impossible by (2). So Ab 5 ;.
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We prove that there exists (y, z) 2 Ab such that y 2 R(a) and z 2 L(a). Then we have b 6 yz 2 R(a)L(a), and b 2 (R(a)L(a)]. Suppose for each (y, z) 2 Ab we have y 62 R(a) or z 62 L(a). Then minffRðaÞ ðyÞ; fLðaÞ ðzÞg ¼ 0 8ðy; zÞ 2 Ab .
ð3Þ
Indeed: Let (y, z) 2 Ab. If y 62 R(a), then fR(a)(y) :¼ 0. Since z 2 S, we have fL(a)(z) P 0. Thus min{fR(a)(y), fL(a)(z)} = 0. If z 62 L(a), then fL(a)(z) :¼ 0. Since y 2 S, we get fR(a)(y) P W 0. Hence we have min{fR(a)(y), fL(a)(z)} = 0. By (3), we have minffRðaÞ ðyÞ; fLðaÞ ðzÞg ¼ 0. ðy;zÞ2Ab
Since Ab 5 ;,we have ðfRðaÞ fLðaÞ ÞðbÞ :¼
_
minffRðaÞ ðyÞ; fLðaÞ ðzÞg.
ðy;zÞ2Ab
Then we have (fR(a) fL(a))(b) = 0, which is impossible by (2).
h
By Proposition 3 and Theorem 1, we have the following: Corollary 1. An ordered semigroup S is regular if and only if for every fuzzy right ideal f and every fuzzy subset g of S, we have f ^ g f g. By Proposition 4 and Theorem 1, we have the following: Corollary 2. An ordered semigroup S is regular if and only if for every fuzzy subset f and every fuzzy left ideal g of S, we have f ^ g f g. Remark 1. The condition ‘‘fR ^ fL fR fL (resp. fR ^ fL = fR fL) for each right ideal R and each left ideal L’’, also characterizes the regular ordered semigroups.
3. Characterization of regular ordered semigroups in terms of fuzzy right, fuzzy left and fuzzy quasi-ideals It is well known that a semigroup (resp. ordered semigroup) S is regular if and only if the right ideals and the left ideals of S are idempotent, and for each right ideal R and each left ideal L of S, the product RL (resp. (RL]) is a quasiideal of S [3,16]. In case of ordered semigroups a right or left ideal A is called idempotent if A = (A2]. It is also known that an le-semigroup S is regular if and only if the right ideal elements and the left ideal elements of S are idempotent, and for every right ideal element a and every left ideal element b of S, the element ab is a quasi-ideal element of S [9]. In this paragraph we show that
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an ordered semigroup S is regular if and only if the fuzzy right and the fuzzy left ideals of S are idempotent and for each fuzzy right ideal f and each fuzzy left ideal g of S, the fuzzy set f g is a fuzzy quasi-ideal of S. Proposition 5. Let S be a groupoid and A, B S. Then we have A B if and only if f A fB : The proof of the proposition is easy. Proposition 6. If S is a groupoid and A, B S, then we have A ¼ B if and only if f A ¼ fB : Proposition 7. Let S be a groupoid and {Aiji 2 I} a family of subsets of S. Then we have ^ fAi ¼ fT Ai . i2I
i2I
T V Proof. Let x 2 S. Then ð i2I fAi ÞðxÞ ¼ ðfT Ai ÞðxÞ. Indeed: If x 2 i2I Ai , then i2I ðfT Ai ÞðxÞ :¼ 1. Since x 2 Ai for each i 2 I, we have fAi ðxÞ :¼ 1 for each T V i2I i 2 I, so ð i2I fAi ÞðxÞ ¼ 1. Let x 62 i2I Ai . Then ðfT Ai ÞðxÞ :¼ 0. Suppose x 62 Aj i2I for some j 2 I. Then infffAi ðxÞji 2 Ig 6 fAj ðxÞ ¼ 0. On the other hand, 0 6 fAi ðxÞ for every i 2 I, from which 0 6 infffAi ðxÞji 2 Ig. Thus we have 0 ¼ infffAi ðxÞji 2 Ig ¼
^
! fAi ðxÞ.
i2I
By Proposition 7, we clearly have the following: Corollary 3. Let S be a groupoid and A, B subsets of S. Then we have fA ^ fB ¼ fA\B. Proposition 8. Let (S, Æ, 6) be an ordered groupoid and A,B S. Then fA fB ¼ fðAB . Proof. Let x 2 S. Then ðfA fB ÞðxÞ ¼ fðAB ðxÞ.
ð4Þ
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Indeed: Let x 2 (AB]. Then f(AB](x) :¼ 1. Since x 6 ab for some a 2 A and b 2 B, we have (a, b) 2 Ax, and Ax 5 ;. Then _ minffA ðyÞ; fB ðzÞg P minffA ðyÞ; fB ðzÞg 8ðy; zÞ 2 Ax . ðfA fB ÞðxÞ :¼ ðy;zÞ2Ax
Since (a, b) 2 Ax, we have ðfA fB ÞðxÞ P minffA ðaÞ; fB ðbÞg. Since a 2 A, b 2 B, we have fA(a) :¼ 1, fB(b) :¼ 1, then min{fA(a), fB(b)} = 1, and (fA fB)(x) P 1. Since fA fB is a fuzzy subset of S, we have (fA fB)(x) 6 1. Therefore (fA fB)(x) = 1, and (4) is satisfied. Let x 62 (AB]. Then f(AB](x) :¼ 0. If Ax = ;, then (fA fB)(x) :¼ 0, and (fA fB)(x) = f(AB](x). Let Ax 5 ;. Then _ ðfA fB ÞðxÞ :¼ minffA ðyÞ; fB ðzÞg. ð5Þ ðy;zÞ2Ax
We prove that min{fA(y), fB(z)} = 0 for all (y, z) 2 Ax. Then, by (5), (fA fB)(x) = 0, and (4) is satisfied. Let (y, z) 2 Ax. Then min{fA(y), ,fB(z)} = 0. Indeed: Since (y, z) 2 Ax, we have x 6 yz. If y 2 A and z 2 B, then yz 2 AB, so x 2 (AB], which is impossible. Thus we have y 62 A or z 62 B. If y 62 A, then fA(y) :¼ 0. Since fB(z) P 0, we have min {fA(y), fB(z)} = 0. If z 62 B then, as in the previous case, we have min{fA(y), fB(z)} = 0. h A non-empty subset Q of an ordered groupoid S is called a quasi-ideal of S if (1) (QS] \ (SQ] Q, (2) If a 2 Q and S 3 b 6 a, then b 2 Q [14]. For an ordered groupoid S, the fuzzy subsets ‘‘0’’ and ‘‘1’’ of S are defined as follows: 0: S ! [0, 1]jx ! 0(x) :¼ 0, 1: S ! [0, 1]jx ! 1(x) :¼ 1. Clearly, the fuzzy subset 0 (resp. 1) of S is the least (resp. the greatest) element of the ordered set (F(S), ) (that is, 0 f and f 1 for every f 2 F(S)). The fuzzy set 0 is the zero element of (F(S), , ) (that is, f 0 = 0 f = 0 and 0 f for every f 2 F(S) [2,23]). Moreover, fS = 1 and f; = 0. Definition 1. Let (S, Æ, 6) be an ordered groupoid. A fuzzy subset f of S is called a fuzzy quasi-ideal of S if
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(1) (f 1) ^ (1 f) f. (2) If x, y 2 S, x 6 y, then f(x) P f(y). Proposition 9. Let (S, Æ. 6) be an ordered groupoid and ; 5 Q S. Then Q is a quasi-ideal of S if and only if fQ is a fuzzy quasi-ideal of S. Proof. ). First of all, we have (fQ 1) ^ (1 fQ) fQ. Indeed: Since 1 = fS, we have ðfQ 1Þ ^ ð1 fQ Þ ¼ ðfQ fS Þ ^ ðfS fQ Þ ¼ fðQS ^ fðSQ (by Proposition 8) ¼ fðQS\ðSQ
ðby Corollary 3Þ.
Since Q is a quasi-ideal of S, we have (QS] \ (SQ] Q. Then, by Proposition 5, we have f(QS]\(SQ] fQ. Hence we have (fQ 1) ^ (1 fQ) fQ. Let now x, y 2 S, x 6 y. Then fQ(x) P fQ(y). Indeed: If y 62 Q, then fQ(y) :¼ 0. Since fQ is a fuzzy subset of S, fQ(x) P 0. Thus fQ(x) P fQ(y). If y 2 Q, then fQ(y) :¼ 1. Since S 3 x 6 y 2 Q and Q is a quasi-ideal of S, we have x 2 Q, then fQ(x) :¼ 1, so fQ(x) P fQ(y). (. We have (QS] \ (SQ] Q. Indeed: Since fQ is a fuzzy quasi-ideal of S, we have (fQ 1) ^ (1 fQ) fQ, then (fQ fS) ^ (fS fQ) fQ. By Proposition 8, f(QS] ^ f(SQ] fQ. By Corollary 3, f(QS]\(SQ] fQ. By Proposition 5, (QS] \ (SQ] Q. Let now x 2 S and x 6 y 2 Q. Then x 2 Q. Indeed: Since x 6 y and fQ is a fuzzy subset of S, we have fQ(x) P fQ(y). Since y 2 Q, we have fQ(y) :¼ 1. So fQ(x) P 1. Since fQ is a fuzzy subset of S, we have fQ(x) 6 1. Thus fQ(x) = 1, and x 2 Q. h Proposition 10. Let (S, Æ, 6) be an ordered groupoid and f (resp. g) a fuzzy right (resp. left) ideal of S. Then f 1 f (resp. 1 g g). Proof. Let f be a fuzzy right ideal of S. First of all, 1 2 F(S). Let a 2 S. Then (f 1)(a) 6 f(a). Indeed: If Aa = ;, then (f 1)(a) :¼ 0. Since f is a fuzzy subset of S, we have f(a) P 0. So (f 1)(a) 6 f(a). Let Aa 5 ;. Then _ ðf 1ÞðaÞ :¼ minff ðyÞ; 1ðzÞg. ðy;zÞ2Aa
We have minff ðyÞ; 1ðzÞg 6 f ðaÞ
8ðy; zÞ 2 Aa .
Indeed: Let (y, z) 2 Aa. Since a 6 yz and f is a fuzzy right ideal of S, we have f(a) P f(yz) P f(y). Since f is a fuzzy set in S, we have f(y) 6 1. Since
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1(z) :¼ 1, we have min{f(y), 1(z)} = f(y) 6 f(a). Hence we have (f 1)(a) 6 f(a). Similarly we prove that 1 g g. h Proposition 11. Let S be an ordered groupoid and f (resp. g) a fuzzy right (resp. fuzzy left) ideal of S. Then f f f (resp. g g g). Proof. Let f be a fuzzy right ideal of S. Since f 1 and f f, by Proposition 1, we have f f f 1. Since f is a fuzzy right ideal of S, by Proposition 10, we have f 1 f. Thus we have f f f. In a similar way, for each fuzzy left ideal g of S, we have g g g. h Proposition 12. Let (S, Æ, 6) be a regular ordered semigroup and f (resp. g) a fuzzy right (resp. fuzzy left) ideal of S. Then f f f (resp. g g g). Proof. Let a 2 S. Then f(a) 6 (f f)(a). Indeed: Since S is regular, there exists x 2 S such that a 6 axa. Then (ax, a) 2 Aa. Since Aa 5 ;, we have _ minff ðyÞ; f ðzÞg P minff ðyÞ; f ðzÞg 8ðy; zÞ 2 Aa . ðf f ÞðaÞ :¼ ðy;zÞ2Aa
Since (ax, a) 2 Aa, we get (f f)(a) P min{f(ax), f(a)}. Since a 6 axa and f is a fuzzy right ideal of S, we have f ðaÞ P f ððaxÞaÞ P f ðaxÞ P f ðaÞ. Hence we have f(ax) = f(a), so min{f(ax), f(a)} = f(a), and (f f)(a) P f(a). Similarly, for each fuzzy left ideal g of S, we get g g g. h Definition 2. A fuzzy subset f of S is called idempotent if f f = f. By Propositions 11 and 12 we have the following: Proposition 13. If S is a regular ordered semigroup, then the fuzzy right and the fuzzy left ideals of S are idempotent. An ordered groupoid in which the right (resp. left) ideals are ideals (i.e. twosided ideals) is called a duo (cf. [11,13,15]). Lajos was arguable the first who introduced the concept of a duo semigroup (cf. e.g. [32]). The commutative semigroups (resp. ordered semigroups) are clearly duo. By Proposition 13 and Theorem 1 (or just by Theorem 1), we have the following: Corollary 4. A duo ordered semigroup S is regular if and only if the fuzzy ideals of S are idempotent.
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Proof. (. Let f be a fuzzy right ideal and g a fuzzy left ideal of S. Since S is a duo, the f, g are fuzzy ideals of S. Then f ^ g is a fuzzy ideal of S and, by hypothesis, we have f ^ g ¼ ðf ^ gÞ ðf ^ gÞ ðf f Þ ^ ðf gÞ ^ ðg f Þ ^ ðg gÞ f g. Then, by Theorem 1, S is regular. The rest of the proof is an immediate consequence of Proposition 13 (or Theorem 1). h Proposition 14. Let (S, Æ, 6) be a regular ordered semigroup, f a fuzzy right ideal of S and g a fuzzy left ideal of S. Then f g is a fuzzy quasi-ideal of S. Proof. By Theorem 1, we have f ^ g = f g. On the other hand, f ^ g is a fuzzy quasi-ideal of S, that is, ((f ^ g) 1) ^ (1 (f ^ g)) f ^ g. Indeed: Since f ^ g f, g, by Proposition 1, we have (f ^ g) 1 f 1, g 1. As we have already seen, (F(S), ) is a lattice. Thus, by the last inequalities, we have ðf ^ gÞ 1 ðf 1Þ ^ ðg 1Þ f 1. Since f is a fuzzy right ideal of S, by Proposition 10, we have f 1 f. Thus we have (f ^ g) 1 f. Similarly we prove that 1 (f ^ g) g. Hence we have ((f ^ g) 1) ^ (1 (f ^ g)) f ^ g. Let now x, y 2 S, x 6 y. Then (f ^ g)(x) P (f ^ g)(y). Indeed: We have (f ^ g)(x) :¼ min{f(x), g(x)}. Since x 6 y and f is a fuzzy right ideal of S, we have f(x) P f(y). Similarly we get g(x) P g(y). Then minff ðxÞ; gðxÞg P minff ðyÞ; gðyÞg; so ðf ^ gÞðxÞ P minff ðyÞ; gðyÞg :¼ ðf ^ gÞðyÞ.
A subset A of an ordered groupoid is called idempotent if A = (A2] [16]. Lemma 4 [16]. An ordered semigroup S is regular if and only if the right and the left ideals of S are idempotent and for every right ideal R and every left ideal L of S, the set (RL] is a quasi-ideal of S. Proposition 15. Let S be an ordered groupoid. A fuzzy right (resp. left) ideal f of S is idempotent if and only if f f f (resp. g g g). Proof. ). Let f = f f. Since ‘‘’’ is an order on F(S), we have (f, f f) 2 , that is, f f f. (. Let f f f. Since f f and f 1, we have f f f 1 f. Thus f = f f. The rest of the prove is similar. h
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Proposition 16. Let S be an ordered semigroup such that for every right ideal R and every left ideal L of S the fuzzy subsets fR and fL of S are idempotent and the product fR fL is a fuzzy quasi-ideal of S. Then S is regular. Proof. Let R be a right ideal of S. Then R is idempotent. Indeed: Since R is a right ideal of S, by Lemma 3, the characteristic function fR is a fuzzy right ideal of S. By hypothesis, fR = fR fR. By Proposition 8, fR fR ¼ fðR2 . So we have fR ¼ fðR2 and, by Proposition 6, R = (R2]. In a similar way we prove that the left ideals of S are idempotent. Let now R be a right ideal and L a left ideal of S. Then (RL] is a quasi-ideal of S. Indeed: By hypothesis, fR fL is a fuzzy quasi-ideal of S. By Proposition 8, fR fL = f(RL]. Since f(RL] is a fuzzy quasiideal of S, by Proposition 9, (RL] is a guasi-ideal of S. Hence, by Lemma 4, S is regular. h By Lemmas 2, 3 and Proposition 16, we have the following: Proposition 17. Let S be an ordered semigroup for which the fuzzy right and the fuzzy left ideals of S are idempotent and for each fuzzy right ideal f and each fuzzy left ideal g of S the product f g is a fuzzy quasi-ideal of S. Then S is regular. By Propositions 13, 14 and 17, we have the following: Theorem 2. An ordered semigroup S is regular if and only if the fuzzy right and the fuzzy left ideals of S are idempotent and for each fuzzy right ideal f and each fuzzy left ideal g of S, the fuzzy set f g is a fuzzy quasi-ideal of S. Remark 2. Each of the following three conditions also characterizes the regular ordered semigroups. (1) ‘‘For each fuzzy right ideal f and each fuzzy left ideal g of S, we have f f f, g g g and f g is a fuzzy quasi-ideal of S.’’ (2) ‘‘For each right ideal R and each left ideal L of S, we have fR = fR fR, fL = fL fL and fR fL is a fuzzy quasi-ideal of S.’’ (3) ‘‘For each right ideal R and each left ideal L of S, we have fR fR fR, fL fL fL and fR fL is a fuzzy quasi-ideal of S.’’ Acknowledgements We express our warmest thanks to the Editor-in-Chief of the journal Professor Witold Pedrycz for editing, communicating the paper, and his useful comments. We also express our warmest thanks to the referee for his interest in our
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work and his time to read the manuscript very carefully and his useful comments. He did the best as a referee. This research has been supported by the Special Research Account of the University of Athens (grant no. 70/4/5630).
References [1] A. Bargiela, W. Pedrycz, Granular Computing. An Introduction, The Kluwer International Series in Engineering and Computer Science, vol. 717, Kluwer Academic Publishers, Boston, MA, 2003, ISBN 1-4020-7273-2, xx + 452 pp. [2] G. Birkhoff, Lattice Theory, Amer. Math. Soc., Coll. Publ., Providence, Rhode Island, 1967. [3] J. Calais, Demi-groupes quasi-inversifs, C.R. Acad. Sci. Paris 252 (1961) 2357–2359. [4] C.L. Chang, Fuzzy topological spaces, J. Math. Anal. Appl. 24 (1968) 182–190. [5] P. Ekel, W. Pedrycz, R. Schinzinger, A general approach to solving a wide class of fuzzy optimization problems, Fuzzy Sets Syst. 97 (1) (1998) 49–66. [6] K. Ise´ki, A characterization of regular semigroups, Proc. Japan Acad. 32 (1956) 676–677. [7] N. Kehayopulu, On intra-regular _e-semigroups, Semigroup Forum 19 (1980) 111–121. [8] N. Kehayopulu, On regular poe-semigroups, in: G. Eigenthaler, H.K. Kaiser, W.B. Mu¨ller, W. No¨bauer (Eds.), Contributions to General Algebra 2, Proceedings of the Klagenfurt Conf., 10–13 June 1982, Verlag Ho¨lder–Pichler–Tempsky, Wien/Verlag B.G. Teubner, Stuttgart, 1983, pp. 191–196. [9] N. Kehayopulu, On regular le-semigroups, Semigroup Forum 28 (1984) 371–372. [10] N. Kehayopulu, On weakly prime ideals of ordered semigroups, Math. Japon. 35 (6) (1990) 1051–1056. [11] N. Kehayopulu, On right regular and right duo ordered semigroups, Math. Japon. 36 (2) (1991) 201–206. [12] N. Kehayopulu, On regular duo ordered semigroups, Math. Japon. 37 (3) (1992) 535–540. [13] N. Kehayopulu, On left regular and left duo poe-semigroups, Semigroup Forum 44 (1992) 306–313. [14] N. Kehayopulu, On regular, intra-regular ordered semigroups, Pure Math. Appl. 4 (4) (1993) 447–461. [15] N. Kehayopulu, On regular, regular duo ordered semigroups, Pure Math. Appl. 5 (2) (1994) 161–176. [16] N. Kehayopulu, On regular ordered semigroups, Math. Japon. 45 (3) (1997) 549–553. [17] N. Kehayopulu, M. Tsingelis, A note on fuzzy sets in semigroups, Sci. Math. 2 (3) (1999) 411– 413. [18] N. Kehayopulu, M. Tsingelis, A characterization of weakly prime and weakly semiprime ideals of semigroups in terms of fuzzy subsets, Sovremennaja Algebra 4 (24) (1999) 85–88. [19] N. Kehayopulu, M. Tsingelis, A note on fuzzy sets of groupoids-semigroups, Sci. Math. 3 (2) (2000) 247–250. [20] N. Kehayopulu, X.-Y. Xie, M. Tsingelis, A characterization of prime and semiprime ideals of semigroups in terms of fuzzy subsets, Soochow J. Math. 27 (2) (2001) 139–144. [21] N. Kehayopulu, M. Tsingelis, Fuzzy sets in ordered groupoids, Semigroup Forum 65 (2002) 128–132. [22] N. Kehayopulu, M. Tsingelis, On semigroups which are groups, J. Fuzzy Math., Int. Fuzzy Math. Inst. 11 (3) (2003) 751–755. [23] N. Kehayopulu, M. Tsingelis, The embedding of an ordered groupoid into a poe-groupoid in terms of fuzzy sets, Inform. Sci. 152 (2003) 231–236.
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[24] N. Kehayopulu, M. Tsingelis, Fuzzy bi-ideals in ordered semigroups, Inform. Sci. 171 (2004) 13–28. [25] L. Kova´cs, A note on regular rings, Publ. Math. Debrecen 4 (1956) 465–468. [26] N. Kuroki, Fuzzy bi-ideals in semigroups, Comment. Math. Univ. St. Pauli XXVIII-1 (1979) 17–21. [27] N. Kuroki, On fuzzy ideals and fuzzy bi-ideals in semigroups, Fuzzy Sets Syst. 5 (1981) 203– 215. [28] N. Kuroki, Fuzzy semiprime ideals in semigroups, Fuzzy Sets Syst. 8 (1982) 71–79. [29] N. Kuroki, On fuzzy semigroups, Inform. Sci. 53 (1991) 203–236. [30] N. Kuroki, Fuzzy generalized bi-ideals in semigroups, Inform. Sci. 66 (1992) 235–243. [31] N. Kuroki, Fuzzy semiprime quasi-ideals in semigroups, Inform. Sci. 75 (1993) 201–211. [32] S. Lajos, On semigroups which are semilattices of groups, Karl Marx Univ. Econ., Dept. Math., Budapest, vol. 2, 1971, i + 14 pp. [33] W.-j. Liu, Fuzzy invariant subgroups and fuzzy ideals, Fuzzy Sets Syst. 8 (1982) 133–139. [34] J.N. Mordeson, D.S. Malik, Fuzzy automata and languages, Theory and Applications, Computational Mathematics Series, Chapman and Hall/CRC, Boca Raton, FL, 2002, ISBN 158488-225-5, xx + 556 pp. [35] J.N. Mordeson, D.S. Malik, N. Kuroki, Fuzzy Semigroups, Studies in Fuzziness and Soft Computing, vol. 131, Springer-Verlag, Berlin, 2003, ISBN 3-540-03243-6, x + 319 pp. [36] J. von Neumann, On regular rings, Proc. Nat. Acad. Sci. USA 22 (1936) 707–713. [37] H.T. Nguyen, N.R. Prasad, C.L. Walker, E.A. Walker, A First Course in Fuzzy and Neural Control, Chapman and Hall/CRC, Boca Raton, FL, 2003, ISBN 1-58488-244-1, x + 301 pp. [38] W. Pedrycz, F. Gomide, An Introduction to Fuzzy Sets, Analysis and Design. With a foreword by Lotfi A. Zadeh. Complex Adaptive Systems. A Bradford Book, MIT Press, Cambridge, MA, 1998, ISBN 0-262-16171-0, xxiv + 465 pp. [39] C. Rickart, General Theory of Banach Algebras, Van Nostrand, Princeton, 1960. [40] A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl. 35 (1971) 512–517. [41] L.A. Zadeh, Fuzzy sets, Inform. Control 8 (1965) 338–353. [42] L.A. Zadeh, Fuzzy Sets and Applications Selected papers. Edited and with a preface by R.R. Yager, R.M. Tong, S. Ovchinnikov and H.T. Nguyen, A Wiley-Interscience Publication, John Wiley and Sons Inc., New York, 1987, ISBN 0-471-85710-6, x + 684 pp. [43] L.A. Zadeh, Fuzzy Sets, Fuzzy Logic, and Fuzzy Systems Selected papers by Lotfi A. Zadeh, Edited and with a preface by George J. Klir and Bo Yuan, Advances in Fuzzy Systems— Applications and Theory, vol. 6, World Scientific Publishing Co., Inc., River Edge, NJ, 1996, ISBN 981-02-2421-4; 981-02-2422-2, xiv + 826 pp. [44] L.A. Zadeh, Probability theory should be based on fuzzy logic—a contentious view, Soft Methodology and Random Informations Systems, Adv. Soft Comput., vols. 3–7, Springer, Berlin, 2004. [45] L.A. Zadeh, Toward a generalized theory of uncertainty (GTU)—an outline, Inform. Sci. 172 (1–2) (2005) 1–40. [46] H.-J. Zimmermann, Fuzzy Set Theory and Its Applications. With a foreword by L.A. Zadeh. International Series in Management Science/Operation Research, Kluwer–Nijhoff Publishing, Boston, MA, 1985, ISBN 0-89838-150-9, xii+363 pp. [47] H.-J. Zimmermann, Fuzzy Set Theory and Its Applications. With a foreword by L.A. Zadeh, fourth ed., Kluwer Academic Publishers, Boston, MA, 2001, ISBN 0-7923-7435-5, XXVi + 514 pp.
Regular ordered semigroups in terms of fuzzy subsets Niovi Kehayopulu *, Michael Tsingelis Department of Mathematics, University of Athens, Panepistimiopolis, 157 84 Athens, Greece Received 12 July 2005; received in revised form 28 December 2005; accepted 1 February 2006
Abstract Given a set S, a fuzzy subset of S (or a fuzzy set in S) is, by definition, an arbitrary mapping f : S ! [0, 1] where [0, 1] is the usual interval of real numbers. If the set S bears some structure, one may distinguish some fuzzy subsets of S in terms of that additional structure. This important concept of a fuzzy set was first introduced by Zadeh. Fuzzy groups have been first considered by Rosenfeld, fuzzy semigroups by Kuroki. A theory of fuzzy sets on ordered groupoids and ordered semigroups can be developed. Some results on ordered groupoids–semigroups have been already given by the same authors in [N. Kehayopulu, M. Tsingelis, Fuzzy sets in ordered groupoids, Semigroup Forum 65 (2002) 128–132; N. Kehayopulu, M. Tsingelis, The embedding of an ordered groupoid into a poe-groupoid in terms of fuzzy sets, Inform. Sci. 152 (2003) 231–236; N. Kehayopulu, M. Tsingelis, Fuzzy bi-ideals in ordered semigroups, Inform. Sci. 171 (2004) 13–28] where S has been endowed with the structure of an ordered semigroup and defined ‘‘fuzzy’’ analogous for several notions that have been proved to be useful in the theory of ordered semigroups. The characterization of regular rings in terms of right and left ideals is well known. The characterization of regular semigroups and regular ordered semigroups in terms of left and right ideals or in terms of left, right ideals
* Corresponding author. Address: Nikomidias 18, 161 22 Kesariani, Greece. Tel.: +30 2107249762. E-mail address: [email protected] (N. Kehayopulu).
0020-0255/$ - see front matter 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.ins.2006.02.004
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and quasi-ideals is well known as well. The characterization of regular le-semigroups (that is lattice ordered semigroups having a greatest element) in terms of right ideal elements and left ideal elements or right, left and quasi-ideal elements is also known. In the present paper we first give the main theorem which characterizes the regular ordered semigroups by means of fuzzy right and fuzzy left ideals. Then we characterize the regular ordered semigroups in terms of fuzzy right, fuzzy left ideals and fuzzy quasi-ideals. The paper serves as an example to show that one can pass from the theory of ordered semigroups to the theory of ‘‘fuzzy’’ ordered semigroups. Some of our results are true for ordered groupoids in general. 2006 Elsevier Inc. All rights reserved. 2000 Mathematics Subject Classification: 06F05; 06D72; 08A72 Keywords: Ordered groupoid; Ordered semigroup; Fuzzy subset; Regular ordered semigroup; Left (right) ideal; Fuzzy left (right) ideal; Quasi-ideal of an ordered groupoid; Fuzzy quasi-ideal of an ordered groupoid; Idempotent fuzzy set (subset)
1. Introduction and prerequisites The application of fuzzy technology in information precessing is already important and it will certainly increase in importance in the future. Our aim is to promote research and the development of fuzzy technology by studying the fuzzy ordered semigroups. The goal is to explain new methodological developments in fuzzy ordered semigroups which will also be of growing importance in the future. This paper can be a bridge passing from the theory of ordered semigroups to the theory of fuzzy ordered semigroups. According to this paper, fundamental notions of ordered semigroups, like the notion of von Neumann regular ordered semigroups, can be expressed in terms of fuzzy sets, and the corresponding properties, like the important conditions which characterize the regular ordered semigroups, remain true (with natural modifications) in case of ordered semigroups, as well. On the other hand, many results of ordered semigroups generalize results of semigroups (without order), by taking the trivial equality relation as the order of the semigroups. On the other hand, the application of fuzzy semigroups in information processing, like in fuzzy coding, fuzzy languages and fuzzy finite state machines, have been proved to be useful. Given a set S, a fuzzy subset of S (or a fuzzy set in S) is, by definition, an arbitrary mapping f : S ! [0, 1] where [0, 1] is the unit segment of the real line. If the set S bears some structure, one may distinguish some fuzzy subsets of S in terms of that additional structure. In fact, a fuzzy set is a class of objects with a continuum of grades of membership. Such a set is characterized by a membership (characteristic function) which assigns to each object a grade of membership ranging between zero and one [41]. This important concept of the fuzzy set was first introduced by Zadeh in [41]. Since then, many papers on fuzzy sets
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appeared showing the importance of the concept and its applications to logic, set theory, group theory, groupoids, real analysis, measure theory, topology, etc. Many notions of mathematics are extended to such sets, and various properties of these notions in the context of fuzzy sets are established. The concept of a fuzzy set introduced by Zadeh, was applied in [4] to generalize some of the basic concepts of general topology. Rosenfeld [40] was the first who considered the case when S is a groupoid. He gave the definition of a fuzzy subgroupoid and the fuzzy left (right, two-sided) ideal of S and justified these definitions by showing that a (conventional) subset A of a groupoid S is a (conventional) subgroupoid or a left (right, two-sided) ideal of S if the characteristic function fA : S ! ½0; 1jx ! fA ðxÞ :¼
1
if x 2 A
0
if x 62 A
is, respectively, a fuzzy subgroupoid or a fuzzy left (right, two-sided) ideal of S. Kuroki has been first studied the fuzzy sets on semigroups [26–31]. See also Liu’s paper [33] where ‘‘fuzzy’’ analogous of several further important notions, e.g. those of bi-ideals or interior ideals have been defined and justified in a similar fashion. A survey of selected papers by Zadeh have been published in [42], where his main field of interest is centered around linguistic modeling of complex systems, approximate reasoning, and the use of fuzzy logical and natural language tools in expert systems and other areas of man–machine interaction. Selected papers by Zadeh have been also recently published in [43], which contains his deeply influential contributions to the field of fuzzy sets and their applications to approximate reasoning and modeling. New tools are needed for solving more difficult social and biological problems. This type of mathematics will be capable of handling uncertainties, making decisions and modeling very large systems and networks which are complex, non-linear and distributive. For an important article by Zadeh towards a generalized theory of uncertainty we refer to [45]. In one of his recent papers Zadeh introduced a new idea to explore the relationship between probabilities and fuzzy sets (see [44]). One of the areas in which fuzzy sets have been applied most extensively is in modeling for managerial decision making. The fuzzy set theory can be applied to the area of human decision making, as well [46]. Interesting results on fuzzy set theory and its applications have been also published by Zimmermann in [47], where one can see the application of fuzzy technology in information processing which is already important. In [47] one can also find new methodological developments in dynamic fuzzy data analysis, which will also be of growing importance in the future. Granular computing refers to the representation of information in the form of aggregates, called granules. If granules are modeled as fuzzy sets, then fuzzy logics are used. This new computing methodology has been considered by Bargiela and Pedrycz in [1]. A presentation of updated trends in fuzzy set theory and its applications has been considered by Petrycz
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and Gomide in [38]. Results of research into the use of fuzzy sets for handling various forms of uncertainty in the optimal design and control of complex systems are presented by Ekel et al. in [5]. Both fuzzy and neural control technologies are firmly based upon the principles of classical control theory [37]. A systematic exposition of fuzzy semigroups by Mordeson et al. appeared in [35], where one can find theoretical results on fuzzy semigroups and their use in fuzzy coding, fuzzy finite state machines and fuzzy languages. Fuzziness has a natural place in the field of formal languages. The monograph by Mordeson and Malik [34] deals with the application of a fuzzy approach to the concepts of automata and formal languages. This is the first monograph systematically presenting developments in the field. Fuzzy sets on semigroups have been also considered by Kehayopulu et al. in [20] and by Kehayopulu and Tsingelis in [17–19,22]. A theory of fuzzy sets on ordered groupoids and ordered semigroups can be developed. We endow S with the structure of an ordered groupoid or semigroup and define ‘‘fuzzy’’ analogous for several notions that have been proved to be useful in the theory of ordered semigroups. Following the terminology given by Zadeh, if S is an ordered groupoid (resp. ordered semigroup), a fuzzy set in S (or a fuzzy subset of S) is any mapping of S into the real closed interval [0, 1]. Based on the terminology given by Zadeh, fuzzy sets in ordered groupoids have been first considered by Kehayopulu and Tsingelis in [21,23,24]. Moreover, each ordered groupoid can be embedded into an ordered groupoid having a greatest element in terms of fuzzy sets [23]. The concept of regular rings was introduced by von Neumann [36]. It is well known that the class of such rings plays an important role in the abstract algebra, in the theory of Banach algebras [39] and in the continuous geometry [36]. An interesting result is that the set of all transformations on a finite dimensional vector space over a field forms a regular ring. Kova´cs characterized the regular rings as rings satisfying the condition A \ B = AB for every right ideal P A and every left ideal B, where AB is the set of all finite sums of the form aibi; ai 2 A, bi 2 B [25]. Ise´ki studied the same for semigroups. He proved that a semigroup (S, Æ) is regular if and only if for every right ideal A and every left ideal B of S, A \ B = AB [6]. As a consequence, a commutative semigroup S is regular if and only if every ideal of S is idempotent [6]. Calais gave the following characterization of regular semigroups: A semigroup S is regular if and only if the right and the left ideals of S are idempotent and for every right ideal A and every left ideal B of S, the product AB is a quasi-ideal of S [3]. The concepts of l-semigroups (: lattice ordered semigroups) and right–left ideal elements in ordered groupoids are due to Birkhoff [2, p. 323, 328]. An l-semigroup is a semigroup S at the same time a lattice such that a(b _ c) = ab _ ac and (a _ b)c = ac _ bc for every a, b, c 2 S. An le-semigroup is an l-semigroup S having a greatest element ‘‘e’’ (i.e. e P a for each a 2 S). An le-semigroup S is called regular, if for each x 2 S, we have x 6 xex [7]. An le-semigroup S is
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regular if and only if a ^ b = ab, equivalently, a ^ b 6 ab for every right ideal element a and every left ideal element b of S [8]. A commutative le-semigroup S is regular if and only if the ideal elements of S are idempotent (cf. [8, Theorem 2]). An le-semigroup S is regular if and only if for every right ideal element x and every left ideal element y of S, we have x2 = x, y2 = y (which means that, the right and the left ideals of S are idempotent), and the product xy is a quasiideal element of S [9]. It was proved in [16] that an ordered semigroup S is regular if and only if for every right ideal A and every left ideal B of S we have A \ B (AB], equivalently, A \ B = (AB]. Moreover, an ordered semigroup (S, Æ, 6) is regular if and only if the right ideals and the left ideals of S are idempotent and for every right ideal A and every left ideal B of S, the set (AB] is a quasi-ideal of S. As a consequence, a duo and so a commutative ordered semigroup S is regular if and only if the ideals of S are idempotent [16]. The results on semigroups—without order—mentioned above can be also obtained applying either the corresponding results on le-semigroups based on ideal elements or the results on ordered semigroups based on ideals (cf. e.g. [9,16]). As a consequence, the condition ‘‘A \ B AB for each right ideal A and each left ideal B’’ also characterizes the regular semigroups [16]. For further information we refer to [9,16]. In this paper we first give the main theorem which characterizes the regular ordered semigroups in terms of fuzzy right and fuzzy left ideals. We prove that an ordered semigroup S is regular if and only if for every fuzzy right ideal f and every fuzzy left ideal g of S we have f ^ g f g, equivalently f ^ g = f g, where f g is the multiplication defined for fuzzy subsets. As a result, a duo, and so a commutative ordered semigroup is regular if and only if the fuzzy ideals of S are idempotent. Then, we introduce the concept of a fuzzy quasi-ideal of ordered groupoids and the concept of idempotent fuzzy set in ordered groupoids, and we characterize the regular ordered semigroups in terms of fuzzy right ideals, fuzzy left ideals, and fuzzy quasi-ideals. We prove that an ordered semigroup S is regular if and only if the fuzzy right ideals and the fuzzy left ideals of S are idempotent and for each fuzzy right ideal f and each fuzzy left ideal g of S, the product f g is a fuzzy quasi-ideal of S. Thus, in addition with the characterizations of regular ordered semigroups we already have, we obtain equivalent characterizations of regular ordered semigroups using the fuzzy sets. The characterizations we obtain correspond to the characterizations of regular semigroups and regular ordered semigroups due to Kova´cs, Ise´ki, Calais and Kehayopulu. It might be noted that the results on fuzzy groupoids or fuzzy semigroups can be obtained as an application of the corresponding results of fuzzy ordered groupoids or fuzzy ordered semigroups, respectively. Let (S, Æ, 6) be an ordered groupoid. By a fuzzy subset of S we mean a mapping f : S ! [0, 1]. For a 2 S, define Aa :¼ fðy; zÞ 2 S Sja 6 yzg.
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For two fuzzy subsets f and g of S, define 8 W minff ðyÞ; gðzÞg < f g : S ! ½0; 1ja ! ðy;zÞ2Aa : 0
if Aa 6¼ ;; if Aa ¼ ;.
We denote by F(S) the set of all fuzzy subsets of S. One can easily see that the multiplication ‘‘’’ on F(S) is well defined. We define an order relation ‘‘’’ on F(S) as follows: f g if and only if f ðxÞ 6 gðxÞ
8x 2 S.
Let (S, Æ, 6) be an ordered groupoid and {fiji 2 I} a family of fuzzy subsets of S. When we refer W to a ‘‘family’’ V we always assume that it is non-empty. Define the fuzzy subsets i2I fi and i2I fi of S as follows: ! _ _ fi : S ! ½0; 1ja ! fi ðaÞ :¼ supffi ðaÞg and i2I
^
i2I
fi : S ! ½0; 1ja !
i2I
^
i2I
!
fi ðaÞ :¼ inf ffi ðaÞg. i2I
i2I
If I is a finite set, say I = {1, 2, . . . , n}, then clearly n _ i¼1 n ^
fi ðaÞ ¼ maxff1 ðaÞ; f2 ðaÞ; . . . ; fn ðaÞg and fi ðaÞ ¼ minff1 ðaÞ; f2 ðaÞ; . . . ; fn ðaÞg.
i¼1
One can easily prove the following: _ ^ fi ¼ supffi g and fi ¼ inf ffi g. i2I
i2I
i2I
i2I
which means that the set of all fuzzy subset of S, endowed with the order ‘‘’’ defined above, is a complete lattice. For two fuzzy subsets f, g of S, we clearly have f ^ g = min{f, g} and f _ g = max{f, g} (and these are the two operations of the lattice (F(S), )).
2. Characterization of regular ordered semigroups in terms of fuzzy right ideals and fuzzy left ideals The characterization of regular rings, semigroups or ordered semigroups in terms of right ideals and left ideals is well known [25,6,16]. The characterization of regular le-semigroups in terms of right ideal elements and left ideal
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elements is known as well [8]. In this paragraph we give the analogous result by means of fuzzy sets. We prove that an ordered semigroup S is regular if and only if for each fuzzy right ideal f and each fuzzy left ideal g of S, we have f ^ g = f g, equivalently, f ^ g f g. Let (S, Æ, 6) be an ordered groupoid. For A S, we denote ðA :¼ ft 2 Sjt 6 h for some h 2 Ag. We have A (A]. If A B, then (A] (B], (A](B] (AB], ((A]] = (A] [10]. A non-empty subset A of S is called a right (resp. left) ideal of S if (1) AS A (resp. SA A) and (2) If a 2 A and S 3 b 6 a, then b 2 A. If A is both a right and a left ideal of S, then it is called an ideal of S [10]. We denote by R(a) (resp. L(a)) the right (resp. left) ideal of S generated by a (a 2 S). We have R(a) = (a [ aS] and L(a) = (a [ Sa] [10]. An ordered semigroup (S, Æ, 6) is called regular if for every a 2 S there exists x 2 S such that a 6 axa. Equivalent Definitions: (1) A (ASA] "A S. (2) a 2 (aSa] "a 2 S [12]. In particular, if the ordered semigroup S is a poe-semigroup, that is an ordered semigroup having a greatest element ‘‘e’’, then it can be easily seen that S is regular if and only if x 6 xex for all x 2 S. Proposition 1. If (S, Æ, 6) is an ordered groupoid and f1, f2, g1, g2 fuzzy subsets of S such that f1 g1 and f2 g2, then f1 f2 g1 g2. Proof. Let a 2 S. If Aa = ;, then (f1 f2)(a) :¼ 0 6 0 :¼ (g1 g2)(a). Let Aa 5 ;. Then _ ðf1 f2 ÞðaÞ :¼ minff1 ðyÞ; f2 ðzÞg; ðy;zÞ2Aa
ðg1 g2 ÞðaÞ :¼
_
minfg1 ðyÞ; g2 ðzÞg.
ðy;zÞ2Aa
We have minff1 ðyÞ; f2 ðzÞg 6 minfg1 ðyÞ; g2 ðzÞg
8ðy; zÞ 2 Aa .
ð1Þ
Indeed: Let (y,z) 2 Aa. Since y, z 2 S, f1 g1 and f2 g2, we have f1(y) 6 g1(y) and f2(z) 6 g2(z), then minff1 ðyÞ; f2 ðzÞg 6 minfg1 ðyÞ; g2 ðzÞg. By (1), we have _ _ minff1 ðyÞ; f2 ðzÞg 6 minfg1 ðyÞ; g2 ðzÞg; ðy;zÞ2Aa
ðy;zÞ2Aa
thus (f1 f2)(a) 6 (g1 g2)(a). Thus we have f1 f2 g1 g2. h
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By Proposition 1, the set F(S) of all fuzzy subsets of S endowed with the multiplication ‘‘’’ and the order ‘‘’’ is an ordered groupoid. Lemma 1. An ordered semigroup (S, Æ, 6) is regular if and only if for every a 2 S we have R(a) \ L(a) (R(a)L(a)]. Proof. ). Let S be regular. Then for each right ideal A and each subset B of S, we have A \ B (AB]. Indeed: A \ B ððA \ BÞSðA \ BÞ ððASÞB ðAB. Let now a 2 S. Since R(a) is a right ideal of S, we have R(a) \ L(a) (R(a)L(a)]. (. Let a 2 S. Then a 2 RðaÞ \ LðaÞ ðRðaÞLðaÞ ¼ ðða [ aSða [ Sa ððða [ aSÞða [ SaÞ ¼ ðða [ aSÞða [ SaÞ ¼ ða2 [ aSa [ aS 2 a ¼ ða2 [ aSa. Then a 6 a2 or a 6 axa for some x 2 S. Then S is regular.
h
If (S, Æ, 6) is an ordered groupoid and A S, the fuzzy subset fA of S is the characteristic function of A defined as follows: ( 1 if x 2 A; fA : S ! ½0; 1jx ! fA ðxÞ :¼ 0 if x 62 A. Let (S, Æ, 6) be an ordered groupoid. A fuzzy subset f of S is called a fuzzy right ideal of S if (1) f(xy) P f(x) for every x, y 2 S. (2) If x 6 y, then f(x) P f(y). A fuzzy subset f of S is called a fuzzy left ideal of S if (1) f(xy) P f(y) for every x, y 2 S. (2) If x 6 y, then f(x) P f(y) [21]. A fuzzy subset f of S is called a fuzzy ideal of S if it is both a fuzzy right and a fuzzy left ideal of S. One can easily see that a fuzzy subset f is a fuzzy ideal of S if and only if the following assertions hold: (1) f(xy) P max{f(x), f(y)} for every x, y 2 S. (2) If x 6 y, then f(x) P f(y) [21].
Lemma 2 (Cf. [21]). Let S be an ordered groupoid. A non-empty subset L of S is a left ideal of S if and only if the characteristic function fL is a fuzzy left ideal of S.
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In a similar way we prove the following: Lemma 3. Let S be an ordered groupoid. A non-empty subset R of S is a right ideal of S if and only if the characteristic function fR is a fuzzy right ideal of S. Proposition 2. Let (S, Æ, 6) be an ordered groupoid, f a fuzzy right ideal and g a fuzzy left ideal of S. Then f g f ^ g. Proof. Let a 2 S. Then (f g)(a) 6 (f ^ g)(a). In fact: If Aa = ;, then (f g)(a) :¼ 0. Since a 2 S and f ^ g is a fuzzy subset of S, we have (f ^ g)(a) P 0, thus (f g)(a) 6 (f ^ g)(a). Let Aa 5 ;. Then ðf gÞðaÞ :¼
_
minff ðyÞ; gðzÞg.
ðy;zÞ2Aa
We prove that minff ðyÞ; gðzÞg 6 ðf ^ gÞðaÞ
8ðy; zÞ 2 Aa .
Then we have _ minff ðyÞ; gðzÞg 6 ðf ^ gÞðaÞ; ðy;zÞ2Aa
and (f g)(a) 6 (f ^ g)(a). Let now (y, z) 2 Aa. Then min{f(y), g(z)} 6 (f ^ g)(a). Indeed: Since (y, z) 2 Aa, we have y, z 2 S and a 6 yz. Then, since f is a fuzzy right ideal of S, we have f(a) P f(yz) and f(yz) P f(y), so f(a) P f(y). Since g a fuzzy left ideal of S, we have g(a) P g(yz) and g(yz) P g(z), so g(a) P g(z). Then minff ðaÞ; gðaÞg P minff ðyÞ; gðzÞg. So we have ðf ^ gÞðaÞ :¼ minff ðaÞ; gðaÞg P minff ðyÞ; gðzÞg.
Proposition 3. Let S be a regular ordered semigroup. Then for every fuzzy right ideal f and every fuzzy subset g of S, we have f ^ g f g. Proof. Let f be a fuzzy right ideal, g a fuzzy subset of S and a 2 S. Then (f ^ g)(a) 6 (f g)(a). In fact: Since S is regular, there exists x 2 S such that a 6 (ax)a. Then (ax, a) 2 Aa. Since Aa 5 ;, we have _ ðf gÞðaÞ :¼ minff ðyÞ; gðzÞg. ðy;zÞ2Aa
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Besides, (f ^ g)(a) = min{f(a), g(a)}. Since f is a fuzzy right ideal of S, we have f(ax) P f(a). Then min{f(ax), g(a)} P min{f(a), g(a)}. Thus we have ðf ^ gÞðaÞ 6 minff ðaxÞ; gðaÞg. Since (ax, a) 2 Aa, we have minff ðaxÞ; gðaÞg 6
_
minff ðyÞ; gðzÞg.
ðy;zÞ2Aa
Hence we have ðf gÞðaÞ ¼
_
minff ðyÞ; gðzÞg P minff ðaxÞ; gðaÞg P ðf ^ gÞðaÞ.
ðy;zÞ2Aa
Therefore f ^ g f g. h In a similar way we prove the following: Proposition 4. If S is a regular ordered semigroup, then for every fuzzy subset f and every fuzzy left ideal g of S, we have f ^ g f g. Theorem 1. An ordered semigroup S is regular if and only if for every fuzzy right ideal f and every fuzzy left ideal g of S, we have f ^ g f g; equivalently; f ^ g ¼ f g: Proof. Let S be regular, f a fuzzy right ideal and g a fuzzy left ideal of S. By Proposition 3, f ^ g f g. By Proposition 2, f g f ^ g. Then f ^ g = f g. Suppose f ^ g f g for every fuzzy right ideal f and every fuzzy left ideal g of S. Then S is regular. In fact: By Lemma 1, it is enough to prove that RðaÞ \ LðaÞ ðRðaÞLðaÞ
8a 2 S.
Let a 2 S, b 2 R(a) \ L(a). Then b 2 (R(a)L(a)]. Indeed: Since R(a) is a right ideal of S, by Lemma 3, fR(a) is a fuzzy right ideal of S. By Lemma 2, fL(a) is a fuzzy left ideal of S. Then, by hypothesis, ðfRðaÞ ^ fLðaÞ ÞðbÞ 6 ðfRðaÞ fLðaÞ ÞðbÞ. Since (fR(a) ^ fL(a))(b) :¼ min{fR(a)(b), fL(a)(b)}, we have minffRðaÞ ðbÞ; fLðaÞ ðbÞg 6 ðfRðaÞ fLðaÞ ÞðbÞ. Since b 2 R(a) and b 2 L(a), we have fR(a)(b) :¼ 1, fL(a)(b) :¼ 1, then we have min{fR(a)(b), fL(a)(b)} = 1, and 1 6 ðfRðaÞ fLðaÞ ÞðbÞ.
ð2Þ
If Ab = ;, then (fR(a) fL(a))(b) :¼ 0, which is impossible by (2). So Ab 5 ;.
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We prove that there exists (y, z) 2 Ab such that y 2 R(a) and z 2 L(a). Then we have b 6 yz 2 R(a)L(a), and b 2 (R(a)L(a)]. Suppose for each (y, z) 2 Ab we have y 62 R(a) or z 62 L(a). Then minffRðaÞ ðyÞ; fLðaÞ ðzÞg ¼ 0 8ðy; zÞ 2 Ab .
ð3Þ
Indeed: Let (y, z) 2 Ab. If y 62 R(a), then fR(a)(y) :¼ 0. Since z 2 S, we have fL(a)(z) P 0. Thus min{fR(a)(y), fL(a)(z)} = 0. If z 62 L(a), then fL(a)(z) :¼ 0. Since y 2 S, we get fR(a)(y) P W 0. Hence we have min{fR(a)(y), fL(a)(z)} = 0. By (3), we have minffRðaÞ ðyÞ; fLðaÞ ðzÞg ¼ 0. ðy;zÞ2Ab
Since Ab 5 ;,we have ðfRðaÞ fLðaÞ ÞðbÞ :¼
_
minffRðaÞ ðyÞ; fLðaÞ ðzÞg.
ðy;zÞ2Ab
Then we have (fR(a) fL(a))(b) = 0, which is impossible by (2).
h
By Proposition 3 and Theorem 1, we have the following: Corollary 1. An ordered semigroup S is regular if and only if for every fuzzy right ideal f and every fuzzy subset g of S, we have f ^ g f g. By Proposition 4 and Theorem 1, we have the following: Corollary 2. An ordered semigroup S is regular if and only if for every fuzzy subset f and every fuzzy left ideal g of S, we have f ^ g f g. Remark 1. The condition ‘‘fR ^ fL fR fL (resp. fR ^ fL = fR fL) for each right ideal R and each left ideal L’’, also characterizes the regular ordered semigroups.
3. Characterization of regular ordered semigroups in terms of fuzzy right, fuzzy left and fuzzy quasi-ideals It is well known that a semigroup (resp. ordered semigroup) S is regular if and only if the right ideals and the left ideals of S are idempotent, and for each right ideal R and each left ideal L of S, the product RL (resp. (RL]) is a quasiideal of S [3,16]. In case of ordered semigroups a right or left ideal A is called idempotent if A = (A2]. It is also known that an le-semigroup S is regular if and only if the right ideal elements and the left ideal elements of S are idempotent, and for every right ideal element a and every left ideal element b of S, the element ab is a quasi-ideal element of S [9]. In this paragraph we show that
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an ordered semigroup S is regular if and only if the fuzzy right and the fuzzy left ideals of S are idempotent and for each fuzzy right ideal f and each fuzzy left ideal g of S, the fuzzy set f g is a fuzzy quasi-ideal of S. Proposition 5. Let S be a groupoid and A, B S. Then we have A B if and only if f A fB : The proof of the proposition is easy. Proposition 6. If S is a groupoid and A, B S, then we have A ¼ B if and only if f A ¼ fB : Proposition 7. Let S be a groupoid and {Aiji 2 I} a family of subsets of S. Then we have ^ fAi ¼ fT Ai . i2I
i2I
T V Proof. Let x 2 S. Then ð i2I fAi ÞðxÞ ¼ ðfT Ai ÞðxÞ. Indeed: If x 2 i2I Ai , then i2I ðfT Ai ÞðxÞ :¼ 1. Since x 2 Ai for each i 2 I, we have fAi ðxÞ :¼ 1 for each T V i2I i 2 I, so ð i2I fAi ÞðxÞ ¼ 1. Let x 62 i2I Ai . Then ðfT Ai ÞðxÞ :¼ 0. Suppose x 62 Aj i2I for some j 2 I. Then infffAi ðxÞji 2 Ig 6 fAj ðxÞ ¼ 0. On the other hand, 0 6 fAi ðxÞ for every i 2 I, from which 0 6 infffAi ðxÞji 2 Ig. Thus we have 0 ¼ infffAi ðxÞji 2 Ig ¼
^
! fAi ðxÞ.
i2I
By Proposition 7, we clearly have the following: Corollary 3. Let S be a groupoid and A, B subsets of S. Then we have fA ^ fB ¼ fA\B. Proposition 8. Let (S, Æ, 6) be an ordered groupoid and A,B S. Then fA fB ¼ fðAB . Proof. Let x 2 S. Then ðfA fB ÞðxÞ ¼ fðAB ðxÞ.
ð4Þ
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Indeed: Let x 2 (AB]. Then f(AB](x) :¼ 1. Since x 6 ab for some a 2 A and b 2 B, we have (a, b) 2 Ax, and Ax 5 ;. Then _ minffA ðyÞ; fB ðzÞg P minffA ðyÞ; fB ðzÞg 8ðy; zÞ 2 Ax . ðfA fB ÞðxÞ :¼ ðy;zÞ2Ax
Since (a, b) 2 Ax, we have ðfA fB ÞðxÞ P minffA ðaÞ; fB ðbÞg. Since a 2 A, b 2 B, we have fA(a) :¼ 1, fB(b) :¼ 1, then min{fA(a), fB(b)} = 1, and (fA fB)(x) P 1. Since fA fB is a fuzzy subset of S, we have (fA fB)(x) 6 1. Therefore (fA fB)(x) = 1, and (4) is satisfied. Let x 62 (AB]. Then f(AB](x) :¼ 0. If Ax = ;, then (fA fB)(x) :¼ 0, and (fA fB)(x) = f(AB](x). Let Ax 5 ;. Then _ ðfA fB ÞðxÞ :¼ minffA ðyÞ; fB ðzÞg. ð5Þ ðy;zÞ2Ax
We prove that min{fA(y), fB(z)} = 0 for all (y, z) 2 Ax. Then, by (5), (fA fB)(x) = 0, and (4) is satisfied. Let (y, z) 2 Ax. Then min{fA(y), ,fB(z)} = 0. Indeed: Since (y, z) 2 Ax, we have x 6 yz. If y 2 A and z 2 B, then yz 2 AB, so x 2 (AB], which is impossible. Thus we have y 62 A or z 62 B. If y 62 A, then fA(y) :¼ 0. Since fB(z) P 0, we have min {fA(y), fB(z)} = 0. If z 62 B then, as in the previous case, we have min{fA(y), fB(z)} = 0. h A non-empty subset Q of an ordered groupoid S is called a quasi-ideal of S if (1) (QS] \ (SQ] Q, (2) If a 2 Q and S 3 b 6 a, then b 2 Q [14]. For an ordered groupoid S, the fuzzy subsets ‘‘0’’ and ‘‘1’’ of S are defined as follows: 0: S ! [0, 1]jx ! 0(x) :¼ 0, 1: S ! [0, 1]jx ! 1(x) :¼ 1. Clearly, the fuzzy subset 0 (resp. 1) of S is the least (resp. the greatest) element of the ordered set (F(S), ) (that is, 0 f and f 1 for every f 2 F(S)). The fuzzy set 0 is the zero element of (F(S), , ) (that is, f 0 = 0 f = 0 and 0 f for every f 2 F(S) [2,23]). Moreover, fS = 1 and f; = 0. Definition 1. Let (S, Æ, 6) be an ordered groupoid. A fuzzy subset f of S is called a fuzzy quasi-ideal of S if
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(1) (f 1) ^ (1 f) f. (2) If x, y 2 S, x 6 y, then f(x) P f(y). Proposition 9. Let (S, Æ. 6) be an ordered groupoid and ; 5 Q S. Then Q is a quasi-ideal of S if and only if fQ is a fuzzy quasi-ideal of S. Proof. ). First of all, we have (fQ 1) ^ (1 fQ) fQ. Indeed: Since 1 = fS, we have ðfQ 1Þ ^ ð1 fQ Þ ¼ ðfQ fS Þ ^ ðfS fQ Þ ¼ fðQS ^ fðSQ (by Proposition 8) ¼ fðQS\ðSQ
ðby Corollary 3Þ.
Since Q is a quasi-ideal of S, we have (QS] \ (SQ] Q. Then, by Proposition 5, we have f(QS]\(SQ] fQ. Hence we have (fQ 1) ^ (1 fQ) fQ. Let now x, y 2 S, x 6 y. Then fQ(x) P fQ(y). Indeed: If y 62 Q, then fQ(y) :¼ 0. Since fQ is a fuzzy subset of S, fQ(x) P 0. Thus fQ(x) P fQ(y). If y 2 Q, then fQ(y) :¼ 1. Since S 3 x 6 y 2 Q and Q is a quasi-ideal of S, we have x 2 Q, then fQ(x) :¼ 1, so fQ(x) P fQ(y). (. We have (QS] \ (SQ] Q. Indeed: Since fQ is a fuzzy quasi-ideal of S, we have (fQ 1) ^ (1 fQ) fQ, then (fQ fS) ^ (fS fQ) fQ. By Proposition 8, f(QS] ^ f(SQ] fQ. By Corollary 3, f(QS]\(SQ] fQ. By Proposition 5, (QS] \ (SQ] Q. Let now x 2 S and x 6 y 2 Q. Then x 2 Q. Indeed: Since x 6 y and fQ is a fuzzy subset of S, we have fQ(x) P fQ(y). Since y 2 Q, we have fQ(y) :¼ 1. So fQ(x) P 1. Since fQ is a fuzzy subset of S, we have fQ(x) 6 1. Thus fQ(x) = 1, and x 2 Q. h Proposition 10. Let (S, Æ, 6) be an ordered groupoid and f (resp. g) a fuzzy right (resp. left) ideal of S. Then f 1 f (resp. 1 g g). Proof. Let f be a fuzzy right ideal of S. First of all, 1 2 F(S). Let a 2 S. Then (f 1)(a) 6 f(a). Indeed: If Aa = ;, then (f 1)(a) :¼ 0. Since f is a fuzzy subset of S, we have f(a) P 0. So (f 1)(a) 6 f(a). Let Aa 5 ;. Then _ ðf 1ÞðaÞ :¼ minff ðyÞ; 1ðzÞg. ðy;zÞ2Aa
We have minff ðyÞ; 1ðzÞg 6 f ðaÞ
8ðy; zÞ 2 Aa .
Indeed: Let (y, z) 2 Aa. Since a 6 yz and f is a fuzzy right ideal of S, we have f(a) P f(yz) P f(y). Since f is a fuzzy set in S, we have f(y) 6 1. Since
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1(z) :¼ 1, we have min{f(y), 1(z)} = f(y) 6 f(a). Hence we have (f 1)(a) 6 f(a). Similarly we prove that 1 g g. h Proposition 11. Let S be an ordered groupoid and f (resp. g) a fuzzy right (resp. fuzzy left) ideal of S. Then f f f (resp. g g g). Proof. Let f be a fuzzy right ideal of S. Since f 1 and f f, by Proposition 1, we have f f f 1. Since f is a fuzzy right ideal of S, by Proposition 10, we have f 1 f. Thus we have f f f. In a similar way, for each fuzzy left ideal g of S, we have g g g. h Proposition 12. Let (S, Æ, 6) be a regular ordered semigroup and f (resp. g) a fuzzy right (resp. fuzzy left) ideal of S. Then f f f (resp. g g g). Proof. Let a 2 S. Then f(a) 6 (f f)(a). Indeed: Since S is regular, there exists x 2 S such that a 6 axa. Then (ax, a) 2 Aa. Since Aa 5 ;, we have _ minff ðyÞ; f ðzÞg P minff ðyÞ; f ðzÞg 8ðy; zÞ 2 Aa . ðf f ÞðaÞ :¼ ðy;zÞ2Aa
Since (ax, a) 2 Aa, we get (f f)(a) P min{f(ax), f(a)}. Since a 6 axa and f is a fuzzy right ideal of S, we have f ðaÞ P f ððaxÞaÞ P f ðaxÞ P f ðaÞ. Hence we have f(ax) = f(a), so min{f(ax), f(a)} = f(a), and (f f)(a) P f(a). Similarly, for each fuzzy left ideal g of S, we get g g g. h Definition 2. A fuzzy subset f of S is called idempotent if f f = f. By Propositions 11 and 12 we have the following: Proposition 13. If S is a regular ordered semigroup, then the fuzzy right and the fuzzy left ideals of S are idempotent. An ordered groupoid in which the right (resp. left) ideals are ideals (i.e. twosided ideals) is called a duo (cf. [11,13,15]). Lajos was arguable the first who introduced the concept of a duo semigroup (cf. e.g. [32]). The commutative semigroups (resp. ordered semigroups) are clearly duo. By Proposition 13 and Theorem 1 (or just by Theorem 1), we have the following: Corollary 4. A duo ordered semigroup S is regular if and only if the fuzzy ideals of S are idempotent.
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Proof. (. Let f be a fuzzy right ideal and g a fuzzy left ideal of S. Since S is a duo, the f, g are fuzzy ideals of S. Then f ^ g is a fuzzy ideal of S and, by hypothesis, we have f ^ g ¼ ðf ^ gÞ ðf ^ gÞ ðf f Þ ^ ðf gÞ ^ ðg f Þ ^ ðg gÞ f g. Then, by Theorem 1, S is regular. The rest of the proof is an immediate consequence of Proposition 13 (or Theorem 1). h Proposition 14. Let (S, Æ, 6) be a regular ordered semigroup, f a fuzzy right ideal of S and g a fuzzy left ideal of S. Then f g is a fuzzy quasi-ideal of S. Proof. By Theorem 1, we have f ^ g = f g. On the other hand, f ^ g is a fuzzy quasi-ideal of S, that is, ((f ^ g) 1) ^ (1 (f ^ g)) f ^ g. Indeed: Since f ^ g f, g, by Proposition 1, we have (f ^ g) 1 f 1, g 1. As we have already seen, (F(S), ) is a lattice. Thus, by the last inequalities, we have ðf ^ gÞ 1 ðf 1Þ ^ ðg 1Þ f 1. Since f is a fuzzy right ideal of S, by Proposition 10, we have f 1 f. Thus we have (f ^ g) 1 f. Similarly we prove that 1 (f ^ g) g. Hence we have ((f ^ g) 1) ^ (1 (f ^ g)) f ^ g. Let now x, y 2 S, x 6 y. Then (f ^ g)(x) P (f ^ g)(y). Indeed: We have (f ^ g)(x) :¼ min{f(x), g(x)}. Since x 6 y and f is a fuzzy right ideal of S, we have f(x) P f(y). Similarly we get g(x) P g(y). Then minff ðxÞ; gðxÞg P minff ðyÞ; gðyÞg; so ðf ^ gÞðxÞ P minff ðyÞ; gðyÞg :¼ ðf ^ gÞðyÞ.
A subset A of an ordered groupoid is called idempotent if A = (A2] [16]. Lemma 4 [16]. An ordered semigroup S is regular if and only if the right and the left ideals of S are idempotent and for every right ideal R and every left ideal L of S, the set (RL] is a quasi-ideal of S. Proposition 15. Let S be an ordered groupoid. A fuzzy right (resp. left) ideal f of S is idempotent if and only if f f f (resp. g g g). Proof. ). Let f = f f. Since ‘‘’’ is an order on F(S), we have (f, f f) 2 , that is, f f f. (. Let f f f. Since f f and f 1, we have f f f 1 f. Thus f = f f. The rest of the prove is similar. h
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Proposition 16. Let S be an ordered semigroup such that for every right ideal R and every left ideal L of S the fuzzy subsets fR and fL of S are idempotent and the product fR fL is a fuzzy quasi-ideal of S. Then S is regular. Proof. Let R be a right ideal of S. Then R is idempotent. Indeed: Since R is a right ideal of S, by Lemma 3, the characteristic function fR is a fuzzy right ideal of S. By hypothesis, fR = fR fR. By Proposition 8, fR fR ¼ fðR2 . So we have fR ¼ fðR2 and, by Proposition 6, R = (R2]. In a similar way we prove that the left ideals of S are idempotent. Let now R be a right ideal and L a left ideal of S. Then (RL] is a quasi-ideal of S. Indeed: By hypothesis, fR fL is a fuzzy quasi-ideal of S. By Proposition 8, fR fL = f(RL]. Since f(RL] is a fuzzy quasiideal of S, by Proposition 9, (RL] is a guasi-ideal of S. Hence, by Lemma 4, S is regular. h By Lemmas 2, 3 and Proposition 16, we have the following: Proposition 17. Let S be an ordered semigroup for which the fuzzy right and the fuzzy left ideals of S are idempotent and for each fuzzy right ideal f and each fuzzy left ideal g of S the product f g is a fuzzy quasi-ideal of S. Then S is regular. By Propositions 13, 14 and 17, we have the following: Theorem 2. An ordered semigroup S is regular if and only if the fuzzy right and the fuzzy left ideals of S are idempotent and for each fuzzy right ideal f and each fuzzy left ideal g of S, the fuzzy set f g is a fuzzy quasi-ideal of S. Remark 2. Each of the following three conditions also characterizes the regular ordered semigroups. (1) ‘‘For each fuzzy right ideal f and each fuzzy left ideal g of S, we have f f f, g g g and f g is a fuzzy quasi-ideal of S.’’ (2) ‘‘For each right ideal R and each left ideal L of S, we have fR = fR fR, fL = fL fL and fR fL is a fuzzy quasi-ideal of S.’’ (3) ‘‘For each right ideal R and each left ideal L of S, we have fR fR fR, fL fL fL and fR fL is a fuzzy quasi-ideal of S.’’ Acknowledgements We express our warmest thanks to the Editor-in-Chief of the journal Professor Witold Pedrycz for editing, communicating the paper, and his useful comments. We also express our warmest thanks to the referee for his interest in our
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work and his time to read the manuscript very carefully and his useful comments. He did the best as a referee. This research has been supported by the Special Research Account of the University of Athens (grant no. 70/4/5630).
References [1] A. Bargiela, W. Pedrycz, Granular Computing. An Introduction, The Kluwer International Series in Engineering and Computer Science, vol. 717, Kluwer Academic Publishers, Boston, MA, 2003, ISBN 1-4020-7273-2, xx + 452 pp. [2] G. Birkhoff, Lattice Theory, Amer. Math. Soc., Coll. Publ., Providence, Rhode Island, 1967. [3] J. Calais, Demi-groupes quasi-inversifs, C.R. Acad. Sci. Paris 252 (1961) 2357–2359. [4] C.L. Chang, Fuzzy topological spaces, J. Math. Anal. Appl. 24 (1968) 182–190. [5] P. Ekel, W. Pedrycz, R. Schinzinger, A general approach to solving a wide class of fuzzy optimization problems, Fuzzy Sets Syst. 97 (1) (1998) 49–66. [6] K. Ise´ki, A characterization of regular semigroups, Proc. Japan Acad. 32 (1956) 676–677. [7] N. Kehayopulu, On intra-regular _e-semigroups, Semigroup Forum 19 (1980) 111–121. [8] N. Kehayopulu, On regular poe-semigroups, in: G. Eigenthaler, H.K. Kaiser, W.B. Mu¨ller, W. No¨bauer (Eds.), Contributions to General Algebra 2, Proceedings of the Klagenfurt Conf., 10–13 June 1982, Verlag Ho¨lder–Pichler–Tempsky, Wien/Verlag B.G. Teubner, Stuttgart, 1983, pp. 191–196. [9] N. Kehayopulu, On regular le-semigroups, Semigroup Forum 28 (1984) 371–372. [10] N. Kehayopulu, On weakly prime ideals of ordered semigroups, Math. Japon. 35 (6) (1990) 1051–1056. [11] N. Kehayopulu, On right regular and right duo ordered semigroups, Math. Japon. 36 (2) (1991) 201–206. [12] N. Kehayopulu, On regular duo ordered semigroups, Math. Japon. 37 (3) (1992) 535–540. [13] N. Kehayopulu, On left regular and left duo poe-semigroups, Semigroup Forum 44 (1992) 306–313. [14] N. Kehayopulu, On regular, intra-regular ordered semigroups, Pure Math. Appl. 4 (4) (1993) 447–461. [15] N. Kehayopulu, On regular, regular duo ordered semigroups, Pure Math. Appl. 5 (2) (1994) 161–176. [16] N. Kehayopulu, On regular ordered semigroups, Math. Japon. 45 (3) (1997) 549–553. [17] N. Kehayopulu, M. Tsingelis, A note on fuzzy sets in semigroups, Sci. Math. 2 (3) (1999) 411– 413. [18] N. Kehayopulu, M. Tsingelis, A characterization of weakly prime and weakly semiprime ideals of semigroups in terms of fuzzy subsets, Sovremennaja Algebra 4 (24) (1999) 85–88. [19] N. Kehayopulu, M. Tsingelis, A note on fuzzy sets of groupoids-semigroups, Sci. Math. 3 (2) (2000) 247–250. [20] N. Kehayopulu, X.-Y. Xie, M. Tsingelis, A characterization of prime and semiprime ideals of semigroups in terms of fuzzy subsets, Soochow J. Math. 27 (2) (2001) 139–144. [21] N. Kehayopulu, M. Tsingelis, Fuzzy sets in ordered groupoids, Semigroup Forum 65 (2002) 128–132. [22] N. Kehayopulu, M. Tsingelis, On semigroups which are groups, J. Fuzzy Math., Int. Fuzzy Math. Inst. 11 (3) (2003) 751–755. [23] N. Kehayopulu, M. Tsingelis, The embedding of an ordered groupoid into a poe-groupoid in terms of fuzzy sets, Inform. Sci. 152 (2003) 231–236.
N. Kehayopulu, M. Tsingelis / Information Sciences 176 (2006) 3675–3693
3693
[24] N. Kehayopulu, M. Tsingelis, Fuzzy bi-ideals in ordered semigroups, Inform. Sci. 171 (2004) 13–28. [25] L. Kova´cs, A note on regular rings, Publ. Math. Debrecen 4 (1956) 465–468. [26] N. Kuroki, Fuzzy bi-ideals in semigroups, Comment. Math. Univ. St. Pauli XXVIII-1 (1979) 17–21. [27] N. Kuroki, On fuzzy ideals and fuzzy bi-ideals in semigroups, Fuzzy Sets Syst. 5 (1981) 203– 215. [28] N. Kuroki, Fuzzy semiprime ideals in semigroups, Fuzzy Sets Syst. 8 (1982) 71–79. [29] N. Kuroki, On fuzzy semigroups, Inform. Sci. 53 (1991) 203–236. [30] N. Kuroki, Fuzzy generalized bi-ideals in semigroups, Inform. Sci. 66 (1992) 235–243. [31] N. Kuroki, Fuzzy semiprime quasi-ideals in semigroups, Inform. Sci. 75 (1993) 201–211. [32] S. Lajos, On semigroups which are semilattices of groups, Karl Marx Univ. Econ., Dept. Math., Budapest, vol. 2, 1971, i + 14 pp. [33] W.-j. Liu, Fuzzy invariant subgroups and fuzzy ideals, Fuzzy Sets Syst. 8 (1982) 133–139. [34] J.N. Mordeson, D.S. Malik, Fuzzy automata and languages, Theory and Applications, Computational Mathematics Series, Chapman and Hall/CRC, Boca Raton, FL, 2002, ISBN 158488-225-5, xx + 556 pp. [35] J.N. Mordeson, D.S. Malik, N. Kuroki, Fuzzy Semigroups, Studies in Fuzziness and Soft Computing, vol. 131, Springer-Verlag, Berlin, 2003, ISBN 3-540-03243-6, x + 319 pp. [36] J. von Neumann, On regular rings, Proc. Nat. Acad. Sci. USA 22 (1936) 707–713. [37] H.T. Nguyen, N.R. Prasad, C.L. Walker, E.A. Walker, A First Course in Fuzzy and Neural Control, Chapman and Hall/CRC, Boca Raton, FL, 2003, ISBN 1-58488-244-1, x + 301 pp. [38] W. Pedrycz, F. Gomide, An Introduction to Fuzzy Sets, Analysis and Design. With a foreword by Lotfi A. Zadeh. Complex Adaptive Systems. A Bradford Book, MIT Press, Cambridge, MA, 1998, ISBN 0-262-16171-0, xxiv + 465 pp. [39] C. Rickart, General Theory of Banach Algebras, Van Nostrand, Princeton, 1960. [40] A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl. 35 (1971) 512–517. [41] L.A. Zadeh, Fuzzy sets, Inform. Control 8 (1965) 338–353. [42] L.A. Zadeh, Fuzzy Sets and Applications Selected papers. Edited and with a preface by R.R. Yager, R.M. Tong, S. Ovchinnikov and H.T. Nguyen, A Wiley-Interscience Publication, John Wiley and Sons Inc., New York, 1987, ISBN 0-471-85710-6, x + 684 pp. [43] L.A. Zadeh, Fuzzy Sets, Fuzzy Logic, and Fuzzy Systems Selected papers by Lotfi A. Zadeh, Edited and with a preface by George J. Klir and Bo Yuan, Advances in Fuzzy Systems— Applications and Theory, vol. 6, World Scientific Publishing Co., Inc., River Edge, NJ, 1996, ISBN 981-02-2421-4; 981-02-2422-2, xiv + 826 pp. [44] L.A. Zadeh, Probability theory should be based on fuzzy logic—a contentious view, Soft Methodology and Random Informations Systems, Adv. Soft Comput., vols. 3–7, Springer, Berlin, 2004. [45] L.A. Zadeh, Toward a generalized theory of uncertainty (GTU)—an outline, Inform. Sci. 172 (1–2) (2005) 1–40. [46] H.-J. Zimmermann, Fuzzy Set Theory and Its Applications. With a foreword by L.A. Zadeh. International Series in Management Science/Operation Research, Kluwer–Nijhoff Publishing, Boston, MA, 1985, ISBN 0-89838-150-9, xii+363 pp. [47] H.-J. Zimmermann, Fuzzy Set Theory and Its Applications. With a foreword by L.A. Zadeh, fourth ed., Kluwer Academic Publishers, Boston, MA, 2001, ISBN 0-7923-7435-5, XXVi + 514 pp.