Characterizations of ordered semigroups in terms of (∈, ∈ ∨ q)-fuzzy interior ideals

Abstract

In this paper, we give characterizations of ordered semigroups in terms of (∈, ∈ ∨q)-fuzzy interior ideals. We characterize different classes regular (resp. intra-regular, simple and semisimple) ordered semigroups in terms of (∈, ∈ ∨q)-fuzzy interior ideals (resp. (∈, ∈ ∨q)-fuzzy ideals). In this regard, we prove that in regular (resp. intra-regular and semisimple) ordered semigroups the concept of (∈, ∈ ∨q)-fuzzy ideals and (∈, ∈ ∨q)-fuzzy interior ideals coincide. We prove that an ordered semigroup S is simple if and only if it is (∈, ∈ ∨q)-fuzzy simple. We characterize intra-regular (resp. semisimple) ordered semigroups in terms of (∈, ∈ ∨q)-fuzzy ideals (resp. (∈, ∈ ∨q)-fuzzy interior ideals). Finally, we consider the concept of implication-based fuzzy interior ideals in an ordered semigroup, in particular, the implication operators in Lukasiewicz system of continuous-valued logic are discussed.

Ordered Semigroups Based on ∈,∈∨qkδ-Fuzzy Ideals

“…If = { }, then we write ( ] instead of ({ }]. For , ⊆ , we denote [18]. Later, will denote an ordered semigroup, unless otherwise stated.…”

“…Later, will denote an ordered semigroup, unless otherwise stated. For a nonempty subset of , ∈ and ∋ ≤ , then ∈ is called a right (resp., left) ideal [18] of if ⊆ (resp., ⊆ ). If is both a left and a right ideal of then is called a two-sided ideal or simply an ideal of .…”

“…Since then a variety of research has been carried out using this icebreaking idea. More precisely, the aforementioned notion is used by Ma et al [10,11] in 0 -algebras and Davvaz and Mozafar [12] in Lie algebra, and Davvaz and coauthors used the idea of generalized fuzzy sets 2 Advances in Fuzzy Systems in rings [13][14][15], Jun et al [16], Khan and Shabir [17], and Khan et al [18] in ordered semigroups, and Shabir et al [19,20] in semigroups. For further details, the readers are referred to [21][22][23][24][25][26].…”

Generalized Bi-Ideal of Ordered Semigroup Related to Intuitionistic Fuzzy Point

“…In addition, Khan et al [19,20] introduced new forms of intuitionistic fuzzy bi-ideals and intuitionistic fuzzy interior ideals in ordered semigroups. In [21], some characterizations of semigroups in terms of intuitionistic fuzzy interior ideals are provided. In this paper, we define…”

On $(Alpha ,Beta)$-Fuzzy Ideals of Ternary Semigroups

“…Kazanci and Yamak introduced the concept of a generalized fuzzy bi-ideal in semigroups [17] and gave some properties of fuzzy bi-ideals in terms of (∈, ∈ ∨q)-fuzzy bi-ideals. In [19], Khan et al characterized ordered semigroups in terms of (∈, ∈ ∨q)-fuzzy interior ideals and gave some generalized forms of interior ideals in ordered semigroups. Many other researchers used the idea of generalized fuzzy sets and gave several characterizations results in different branches of algebra, for example see [1,15,16,20,22,24,25,28].…”