A study of generalized fuzzy ideals in ordered semigroups (original) (raw)
Related papers
𝒩-Fuzzy Ideals in Ordered Semigroups
International Journal of Mathematics and Mathematical Sciences, 2009
We introduce the concept of N-fuzzy left right ideals in ordered semigroups and characterize ordered semigroups in terms of N-fuzzy left right ideals. We characterize left regular right regular and left simple right simple ordered semigroups in terms of N-fuzzy left N-fuzzy right ideals. The semilattice of left right simple semigroups in terms of N-fuzzy left right ideals is discussed.
Some study of (α, β)-fuzzy ideals in ordered semigroups
2012
Algebraic structures especially an ordered semigroups play a prominent role in mathematics with wide ranging applications in many disciplines such as control engineering, computer arithmetics, coding theory, sequential machines and formal languages. A theory of fuzzy sets in terms of fuzzy points on ordered semigroups can be developed. In this paper, we generalize the concept of (α, β)-fuzzy left (right) ideal of an ordered semigroup S and introduce a new sort of fuzzy left (right) ideals called (∈, ∈ ∨q k)-fuzzy left (right) ideals, where k ∈ [0, 1). In particular, we describe the relationships among ordinary fuzzy ideals and (∈, ∈ ∨q k)fuzzy ideals of an ordered semigroup S. Finally, we characterize regular ordered semigroups in terms of (∈, ∈ ∨q k)-fuzzy left (resp. right) ideals.
New types of fuzzy bi-ideals in ordered semigroups
Neural Computing and Applications, 2012
In Jun et al. (Bull Malays Math Sci Soc 32 :391-408, 2009), (a, b)-fuzzy bi-ideals are introduced and some characterizations are given. In this paper, we generalize the concept of (a, b)-fuzzy bi-ideals and define (2; 2 _q k )-fuzzy bi-ideals in ordered semigroups, which is a generalization of the concept of an (a, b)-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 (2; 2 _q k )-fuzzy bi-ideal and investigate some interesting results of regular and intra-regular ordered semigroups.
Information Sciences, 2014
In this paper we initiated the study of roughness in ordered semigroups based on pseudoorder. We introduced the notions of upper (lower) rough ideal, bi-ideal, prime ideal and also the notions of upper (lower) rough fuzzy ideal, rough prime fuzzy ideal in ordered semigroups. We studied some properties of such ideals. Key words and phrases. Upper (lower) rough ideal, upper (lower) rough bi-ideal, upper (lower) rough fuzzy ideal, upper (lower) rough prime fuzzy ideal. ______________________________________________________________________________ 1. Preliminaries An ordered semigroup () ,, S ⋅≤ , is a poset (,) S ≤ , at the same time a semigroup (,) S ⋅ such that ab ≤ implies axbx ≤ and xa ≤ xb for all ,, abxS ∈. A non-empty subset A of an ordered semigroup S is called a subsemigroup of S if 2 AA ⊆. A non-empty subset I of an ordered semigroup S is called a right (resp. left) ideal of S if ISI ⊆ (resp. SII ⊆) and aI ∈ and bS ∈ such that ba ≤ implies bI ∈. I is called an ideal of S if it is both a right and a left ideal of S. An ideal P of an ordered semigroup S is called a prime ideal if xyP ∈ implies xP ∈ or yP ∈ , for all , xyS ∈ [ ] 8. A subsemigroup B of an ordered semigroup S is called a bi-ideal of S if BSBB ⊆ and aB ∈ and bS ∈ such that ba ≤ implies bB ∈ .
Some New Characterization of Ordered Semigroups in Terms of \((\lambda ,\theta )\)-Fuzzy bi-ideals
International Journal of Algebra and Statistics, 2012
In this paper, we demonstrate a new concept of fuzzy bi-ideals called a (λ, θ)-fuzzy bi-ideals of an ordered semigroup. Fuzzy ideals of type (λ, θ) are the generalization of fuzzy bi-ideals and an (∈, ∈ ∨q)-fuzzy bi-ideals of an ordered semigroup. We show that U(µ, t)( ∅) is a bi-ideal if and only if the fuzzy subset µ is a (λ, θ)-fuzzy i-ideal of S for all t ∈ (λ, θ]. Similarly, A is a bi-ideal if and only if the characteristic function µ A of A is a (λ, θ)-fuzzy bi-ideal of S. With the help of some examples, we show that (λ, θ)-fuzzy bi-ideals ((λ, θ)-fuzzy subsemigroups) are neither fuzzy bi-ideals (fuzzy subsemigroups ) nor (∈, ∈ ∨q)-fuzzy bi-ideals ((∈, ∈ ∨q)-fuzzy subsemigroups) of an ordered semigroup S. Finally, the characterization of completely ordered semigroups in terms of (λ, θ)-fuzzy bi-ideals is given.
Characterizations of regular ordered semigroups in terms of (α,β)-fuzzy generalized bi-ideals
Information Sciences, 2011
In this paper, we introduce the concept of (α, β)-bipolar fuzzy generalized bi-ideal in ordered semigroup, which is a generalization of bipolar-fuzzy generalized bi-ideal in ordered semigroup. Using this concept, we provide some characterization theorems. We prove that in regular ordered semigroup, the concept of (∈, ∈ ∨q)-bipolar fuzzy generalized bi-ideal and (∈, ∈ ∨q)-bipolar fuzzy bi-ideal coincide. We also introduce the upper/lower parts of (∈, ∈ ∨q)-bipolar fuzzy generalized bi-ideals and characterize the regular ordered semigroups in terms of lower part of (∈, ∈ ∨q)-bipolar fuzzy left (resp. right or two sided) ideals and (∈, ∈ ∨q)-bipolar fuzzy generalized bi-ideals.
Characterizations of Ordered Semigroups in Terms of Anti-fuzzy Ideals
Fuzzy Information and Engineering
Adopting the notion of a (k * , q)-quasi-coincidence of a fuzzy point with a fuzzy set, the idea of an (∈ , ∈ ∨(k * , q k))-antifuzzy left (right) ideal, (∈ , ∈ ∨(k * , q k))-antifuzzy ideal and (∈ , ∈ ∨(k * , q k))-antifuzzy (generalized) bi-ideal in ordered semigroups are proposed, that are the generalization of the idea of an antifuzzy left (right) ideal, antifuzzy ideal and antifuzzy (generalized) bi-ideal in ordered semigroups and a few fascinating characterizations are obtained. In this paper, we tend to focus to suggest a connection between standard generalized bi-ideals and (∈ , ∈ ∨(k * , q k))-antifuzzy generalized biideals. In addition, different classes of regular ordered semigroups are characterized by the attributes of this new idea. Finally, the (k * , k)-lower part of an (∈ , ∈ ∨(k * , q k))-antifuzzy generalized bi-ideal is outlined and a few characterizations are mentioned.
Fuzzy Quasi-Ideals of Ordered Semigroups
2009
In this paper, we characterize ordered semigroups in terms of fuzzy quasi% ideals. We characterize left simple, right simple and completely reg% ular ordered semigroups in terms of fuzzy quasi% ideals. We define semi% prime fuzzy quasi% ideal of ordered ...
N-fuzzy quasi-ideals in ordered semigroups
In this paper, we introduce the concept of N-fuzzy quasi-ideals in ordered semigroups and investigate the basic theorem of quasi-ideals of ordered semigroups in terms of N-fuzzy quasi-ideals. We characterize left (resp. right) regular and completely regular ordered semigroups in terms of N-fuzzy quasi-ideals. We dene semiprime Nfuzzy quasi-ideals and characterize completely regular ordered semigroups in terms of semiprime N-fuzzy quasi-ideals. We provide characterizations of some semilattices of left and right simple semigroups in terms of N-fuzzy quasi-ideals.
Characterizations of ordered semigroups by the properties of their fuzzy ideals
Computers & Mathematics with Applications, 2010
A fuzzy subset f of a given set S (or a fuzzy set in S) is, described as an arbitrary function f:S long right arrow [0,1], where [0,1] is the usual closed interval of real numbers. This fundamental concept of a fuzzy set, was first introduced by Zadeh in his pioneering paper [1] of 1965, which ...