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

Abstract

Abstract. In this paper, we introduce the concept of generalized fuzzy ideals in ternary semigroups, which is a generalization of the fuzzy ideals of semigroups. In this regard, we define (α, β)-fuzzy left (right, lateral) ideals, (α, β)-fuzzy quasi-ideals and (α, β)-fuzzy bi-ideals and investigate some related properties of ternary semigroups. Special concentration is paid to (∈, ∈ ∨q )-fuzzy left (right, lateral) ideals, (∈, ∈ ∨q)-fuzzy quasi-ideals and (∈, ∈ ∨q)-fuzzy bi-ideals. Finally, we characterize regular ternary semigroups in terms of these notions.Key words and Phrases:Fuzzy subsets; (∈, ∈ ∨q)-fuzzy left (right, lateral) ideals; (∈, ∈ ∨q)-fuzzy quasi-ideals; (∈, ∈ ∨q)-fuzzy bi-ideals.Abstrak. Dalam makalah ini, kami memperkenalkan konsep ideal fuzzy yang tergeneralisasi dalam semigroup ternary, yang merupakan perluasan dari ideal fuzzy pada semigroup. Dalam hal ini, ideal (α, β)-fuzzy kiri (kanan, lateral), kuasi ideal(α, β)-fuzzy dan bi-ideal (α, β)-fuzzy didefinisikan dan beberapa sifat dari semigroup ternary dibahas. Perhatian khusus ditujukan untuk ideal kiri (kanan, lateral (∈, ∈ ∨q)-fuzzy, kuasi ideal (∈, ∈ ∨q)-fuzzy dan bi-ideal(∈, ∈ ∨q)-fuzzy. Akhirnya, semigrup ternary regular dikarakterisasi berdasarkan sifat-sifat tersebut di atas.Kata kunci: Subhimpunan fuzzy; ideal kiri (∈, ∈ ∨q)-fuzzy (kanan, lateral); kuasi ideal (∈, ∈ ∨q)-fuzzy; bi-ideal (∈, ∈ ∨q)-fuzzy.

Hesitant Fuzzy Sets Approach to Ideal Theory in Ternary Semigroups

“…Let S be a reference set. Then we define HFS on S in terms of a function H that when applied to X returns a subset of [0,1]. For a HFS H on S and x, y ∈ S, we use the notations H x := H(x) and…”

“…Lemma 10. Let H ∈ HF (S) such that H u ⊆ ε for all u ∈ S and [χ (ε, δ)-identity HFS on S. Then,(1) H is a hesitant fuzzy ternary subsemigroup on S if and only if H⋄ H⋄ H ⊑ H. (2) H is a hesitant fuzzy left ideal on S if and only if [χ Proof. (1) Let H be a hesitant fuzzy ternary subsemigroup on S and x ∈ S. Now we have following two cases:…”

A Generalized Definition of Fuzzy Subrings

“…e notions of M-fuzzifying convex structure and (L, M)-fuzzy convex structure are introduced by Shi and Xiu in [9,10]. Actually, fuzzy convexity exists in many mathematical research areas, such as fuzzy vector spaces, fuzzy groups, fuzzy lattices, and fuzzy topologies (see [7,8,[11][12][13][14][15][16][17][18][19][20][21][22][23][24]).…”

Applications of hesitant fuzzy sets to ternary semigroups

“…In 2012, Kar and Sarkar [3] introduced the idea of fuzzy left (right, lateral) ideals of ternary semigroups and provided the characterizations regular and intra-regular ternary semigroups. In 2013, Davvaz et al [4] introduced the concept of generalized fuzzy ideals in ternary semigroups and investigated some of their interesting properties. They defined the ðα; βÞ-fuzzy left (right, lateral) ideals, ðα; βÞ-fuzzy quasi-ideals and ðα; βÞ-fuzzy bi-ideals of ternary semigroups and characterized regular ternary semigroups in terms of ðα;βÞ-fuzzy left (right, lateral) ideals, ðα;βÞ-fuzzy quasi-ideals.…”