Fuzzy Bi-ideals Research Articles

Redefined intuitionistic fuzzy bi-ideals of ordered semigroups

In this article, we try to obtain a more general form than -intuitionistic fuzzy bi-ideals in ordered semigroups. The notion of an -intuitionistic fuzzy bi-ideal is introduced, and several properties are investigated. Characterizations of an -intuitionistic fuzzy bi-ideal are established. A condition for an -intuitionistic fuzzy bi-ideal to be an intuitionistic fuzzy bi-ideal is provided. It is shown that every -intuitionistic fuzzy bi-ideal is an -intuitionistic fuzzy bi-ideal, and every -intuitionistic fuzzy bi-ideal is an -intuitionistic fuzzy bi-ideal but the converse is not true. The important achievement of the study with an -intuitionistic fuzzy bi-ideal is that the notion of an -intuitionistic fuzzy bi-ideal is a special case of an -intuitionistic fuzzy bi-ideal, and, thus, several results in the paper (Jun et al. in Bi-ideals of ordered semigroups based on the intuitionistic fuzzy points (submitted)) are the corollaries of our results obtained in this paper.

Characterizations of semi-groups that are semi-lattices of left groups

Zadeh in 1965 introduce the concept of a fuzzy set which now have a wide range of applications in various fields of Engineering Sciences, Computer Science and Management Sciences. It is worth mentioning here that for applications of fuzzy sets mostly associative algebraic structures are used such as in 2003, Mordeson, Malik and Kuroki have discovered the applications of fuzzy semigroups in fuzzy coding, fuzzy finite-state machines and fuzzy languages. We have applied fuzzy set theory to associative algebraic structures and have discussed related properties. Specifically we have characterized semigroups which are semilattices of left groups by the properties of generalized fuzzy ideals. Key words: -fuzzy ideals, -fuzzy bi-ideals, -fuzzy quasi ideals.

On Interval Valued Fuzzy Bi-Ideals of Semigroups

We introduce the notion of interval-valued fuzzy bi-ideals with respect to the interval Min-norm Min i (briefly, Min i -fuzzy bi-ideals) in semigroups, and we characterize Min i -fuzzy bi-ideals by upper level sets and show that every bi-ideal of a semigroup X can be realized as an upper level ideal of a Min i -fuzzy bi-ideal of X. Finally, we establish the theorems of the homomorphic image and the inverse image.

Ordered semigroups characterized by interval valued (ε, ε $\vee$ q)-fuzzy bi-ideals

In this article, we first introduce the concept of quasi-coincidence of an interval valued fuzzy point with an interval valued fuzzy subset. This concept is a generalization of the concept of quasi-coincidence of a fuzzy point within a fuzzy subset. By using this new idea, we introduce the concept of an interval valued α, β-fuzzy bi-ideals in ordered semigroups. Which is a generalization of the idea of an α, β-fuzzy bi-ideals in ordered semigroups. Some interesting results of the algebraic ordered semigroups in terms of interval valued α, β-fuzzy bi-ideals are given. By using interval valued $e, e $\vee$ q-fuzzy bi-ideals some interesting results are obtained.

More General Forms of $(alpha, beta )$-fuzzy Ideals of Ordered Semigroups

This paper consider the general forms of $(alpha,beta)$-fuzzyleft ideals (right ideals, bi-ideals, interior ideals) of an orderedsemigroup, where$alpha,betain{in_{gamma},q_{delta},in_{gamma}wedgeq_{delta}, in_{gamma}vee q_{delta}}$ and $alphaneqin_{gamma}wedge q_{delta}$. Special attention is paid to$(in_{gamma},ivq)$-left ideals (right ideals, bi-ideals, interiorideals) and some related properties are investigated. Thecharacterization of regular ordered semigroups in terms of$(in_{gamma},ivq)$-fuzzy left (right) ideals,$(in_{gamma},ivq)$-fuzzy bi-ideals and$(in_{gamma},ivq)$-fuzzy interior ideals is also investigated.

Characterization of Intuitionistic Fuzzy Bi-Ideals in \(\Gamma\)-Near Rings

In this paper, the notion of intuitionistic fuzzy bi ideals in \(\Gamma\)-near-rings and maximal intuitionistic fuzzy bi ideals of \(\Gamma\)-near-rings are studied and their properties discussed. Finally, chain conditions relating to intuitionistic fuzzy bi-ideals of \(\Gamma\)-near-rings are investigated.

Some New Characterization of Ordered Semigroups in Terms of \((\lambda ,\theta )\)-Fuzzy bi-ideals

In this paper, we demonstrate a new concept of fuzzy bi-ideals called a \(( \lambda ,\theta ) \)-fuzzy bi-ideals of an ordered semigroup. Fuzzy ideals of type \(( \lambda ,\theta) \) are the generalization of fuzzy bi-ideals and an \(( \in ,\in\vee q) \)-fuzzy bi-ideals of an ordered semigroup. We show that \( U( \mu ,t) ( \neq \emptyset ) \) is a bi-ideal if and only if the fuzzy subset \(\mu\) is a \(( \lambda ,\theta ) \)-fuzzy i-ideal of \(S\) for all \(t\in (\lambda ,\theta ]\). Similarly, \(A\) is a bi-ideal if and only if the characteristic function \(\mu _{A}\) of \(A\) is a \(( \lambda ,\theta ) \)-fuzzy bi-ideal of \(S\). With the� help of some examples, we show that \(( \lambda ,\theta ) \)-fuzzy bi-ideals (\(( \lambda ,\theta ) \)-fuzzy subsemigroups) are neither fuzzy bi-ideals (fuzzy subsemigroups ) nor \(( \in ,\in \vee q) \)-fuzzy bi-ideals (\(( \in ,\in \vee q) \)-fuzzy subsemigroups) of an ordered semigroup \(S\). Finally, the� characterization of completely ordered semigroups in terms of \(( \lambda ,\theta ) \)-fuzzy bi-ideals is given.

Characterizations of fuzzy intra-regular Abel-Grassmann's groupoids

Mostly, models of real life problems are consist of uncertainties. Fuzzy set theory is one of the tool for handling such uncertainties and for applications of fuzzy set theory we have used non-associative algebraic structure namely AG-groupoids. Specifically, we have discussed the -fuzzy left ideals,-fuzzy ideals, -fuzzy semiprime ideals and -fuzzy bi-ideals in an AG-groupoid. We have characterized intra-regular AG-groupoids by the properties of their ideals and -fuzzy ideals. Key words: AG-groupoid, left invertive law, medial law, paramedial law, -fuzzy ideals and -fuzzy bi-ideals.

New types of fuzzy bi-ideals in ordered semigroups

In Jun et al. (Bull Malays Math Sci Soc (2) 32(3):391–408, 2009), (α, β)-fuzzy bi-ideals are introduced and some characterizations are given. In this paper, we generalize the concept of (α, β)-fuzzy bi-ideals and define (\(\in ,\in \vee q_{k}\))-fuzzy bi-ideals in ordered semigroups, which is a generalization of the concept of an (α, β)-fuzzy bi-ideal in an ordered semigroup. Using this concept, some characterization theorems of regular, left (resp. right) regular and completely regular ordered semigroups are provided. In the last section, we give the concept of upper/lower parts of an (\(\in ,\in \vee q_{k}\))-fuzzy bi-ideal and investigate some interesting results of regular and intra-regular ordered semigroups.

A study of fuzzy Abel-Grassmann's groupoids

Fuzzy sets were first introduced by Zadeh in 1965 which have a wide range of applications in various fields of Engineering Sciences, Computer Science and Management Sciences. It is worth mentioning here that for applications of fuzzy sets mostly associative algebraic structures are used such as in 2003, Mordeson, Malik and Kuroki discovered the applications of fuzzy semigroups in fuzzy coding, fuzzy finite-state machines and fuzzy languages. Our aim in this paper is to apply fuzzy sets to non-associative algebraic structures and to develop some new related properties. Specifically we have characterized intra-regular Abel-Grassmann’s groupoids using generalized fuzzy bi-ideals. Key words: Abel-Grassman's groupoid (AG-groupoid), intra-regular, fuzzy sets, fuzzy AG-groupoids, -fuzzy ideals.

Interval-valued Fuzzy Ideals and Bi-ideals of a Semigroup

We apply the concept of interval-valued fuzzy sets to theory of semigroups. We give some properties of interval-valued fuzzy ideals and interval-valued fuzzy bi-ideals, and characterize which is left [right] simple, left [right] duo and a semilattice of left [right] simple semigroups or another type of semigroups in terms of interval-valued fuzzy ideals and intervalvalued fuzzy bi-ideals.

On Fuzzy Ordered Abel-Grassmann's Groupoids

In this paper, we have introduced the concept of fuzzy ordered AG-groupoids which is the generalization of fuzzy ordered semigroups first considered by Kehayopulu and Tsingelis (2002). We have studied some important features of a left regular ordered AG-groupoid interms of fuzzy left ideals, fuzzy right ideals, fuzzy two-sided ideals, fuzzy generalized bi-ideals, fuzzy bi-ideals, fuzzy interior ideals and fuzzy (1,2)-ideals. We have shown that the set of all fuzzy two-sided ideals of a left regular ordered AG-groupoid forms asemilattice structure. We have characterized all the fuzzy ideals of a left regular ordered AG-groupoid. Finally we have characterized a left regular ordered AG-groupoid by their fuzzy left and fuzzy right ideals.

Intra-regular Left Almost Semigroups Characterized by Their Anti Fuzzy Ideals

In this paper, we have introduced anti fuzzy quasi-ideal and anti fuzzy left (right, two-sided) ideals in LA-semigroup. We have proved some results related to anti fuzzy quasi-ideals and anti fuzzy left (right, two-sided) ideals of an LA-semigroup. Further we have characterized an intra-regular LA-semigroup by the properties of their anti fuzzy left (right, two-sided) ideals, anti fuzzy (generalized) bi-ideals and anti fuzzy quasi-ideal.

ON PRIME FUZZY BI-IDEALS OF SEMIGROUPS

In this paper, we introduce and study the prime, strongly prime, semiprime and irreducible fuzzy bi-ideals of a semigroup. We characterize those semigroups for which each fuzzy bi-ideal is semiprime. We also characterize those semigroups for which each fuzzy bi-ideal is strongly prime.

On Anti Fuzzy Ideals in Left Almost Semigroups

In this paper we have studied the concept of Anti fuzzy ideals in Left Almost Semigroups (LA-semigroup in short). The equivalent statement for an LA-semigroup to be a commutative semigroup is proved. The set of all anti fuzzy left ideals, which are idempotents, forms a commutative monoid. Moreover it has been shown that the union of any family of Anti fuzzy left ideals of an LA-semigroup is an anti fuzzy left ideal of F(S). The relation of anti fuzzy left(right) ideals, anti fuzzy interior ideals and anti fuzzy bi-ideals in LA-semigroups has been studied. Anti fuzzy points have been defined in an LA-semigroup and has been shown the representation of largest fuzzy left ideal generated by a fuzzy point.

Characterizations of Semigroups by Their Anti Fuzzy Ideals

In this paper we have defined anti fuzzy interior ideal in semigroups. We characterize regular, intra-regular and left (right) quasi-regular semigroups by the properties of their anti fuzzy ideals, anti fuzzy bi-ideals, anti fuzzy generalized bi-ideals, anti fuzzy interior ideals and anti fuzzy quasi-ideals.

Regular ordered semigroups and intra-regular ordered semigroups in terms of fuzzy subsets

Let $S$ be an ordered semigroup. A fuzzy subset of $S$ is anarbitrary mapping from $S$ into $[0,1]$, where $[0,1]$ is theusual interval of real numbers. In this paper, the concept of fuzzygeneralized bi-ideals of an ordered semigroup $S$ is introduced.Regular ordered semigroups are characterized by means of fuzzy leftideals, fuzzy right ideals and fuzzy (generalized) bi-ideals.Finally, two main theorems which characterize regular orderedsemigroups and intra-regular ordered semigroups in terms of fuzzyleft ideals, fuzzy right ideals, fuzzy bi-ideals or fuzzyquasi-ideals are given. The paper shows that one can pass fromresults in terms of fuzzy subsets in semigroups to orderedsemigroups. The corresponding results of unordered semigroups arealso obtained.

Ordered semigroups characterized by their intuitionistic fuzzy bi-ideals

Fuzzy bi-ideals play an important role in the study of ordered semigroupstructures. The purpose of this paper is to initiate and study theintiuitionistic fuzzy bi-ideals in ordered semigroups and investigate thebasic theorem of intuitionistic fuzzy bi-ideals. To provide thecharacterizations of regular ordered semigroups in terms of intuitionisticfuzzy bi-ideals and to discuss the relationships of left(resp. right andcompletely regular) ordered semigroups in terms intuitionistic fuzzybi-ideals.

Generalized anti fuzzy bi-ideals in ordered semigroups

Using the notion of anti fuzzy points and its besideness to and non-quasi-coincident with a fuzzy set, new concepts of an anti fuzzy bi-ideals in ordered semigroups are introduced and their inter-relations and related properties are investigated.

Characterizations of ordered semigroups by the properties of their fuzzy ideals

In this paper we characterize different classes of ordered semigroups by the properties of their ideals, quasi-ideals and bi-ideals. We also characterize these classes by the properties of their fuzzy ideals, fuzzy quasi-ideals and fuzzy bi-ideals.