Intuitionistic Q-Fuzzy Prime Bi-∧-Ideals of ∧-Semigroups

After the introduction of fuzzy sets by Zadeh [20] in 1965, several researchers have been working on the generalization of the fuzzification. In 1986, Atanassov [2] introduced the notion of intuitionistic fuzzy sets. For more details on the intuitionistic fuzzy sets, we refer to [2-4]. Fuzzy sets are intuitionistic fuzzy sets but the converse is not necessarily true [18]. Kim and Jun [7, 8] introduced the notion of several intuitionistic fuzzy ideals of semigroups. Kim and Lee [9] gave the concept of intuitionistic fuzzy bi-ideals of semigroups. In 1986, Sen and Saha [16] defined the Γ-semigroup and established a relation between regular Γ-semigroup and Γ-group [13, 14]. The notion of bi-Γ-ideals in Γ-semigroup was introduced by Chinram and Jirojkul in [5]. Shabir and Ali [17] introduced the notion of prime bi-ideals of Γ-semigroups. Mustafa et al. [19] introduced the notion of intuitionistic fuzzy Γ-ideals in Γ-semigroups. Sardar et al. [15] gave the concept of intuitionistic fuzzy prime, semiprime ideals and also intuitionistic fuzzy ideal extension in a Γ-semigroups. Lekkoksung [10] defined intuitionistic fuzzy bi-Γ-ideals of Γ-semigroups. In [1] Akram, introduced the notion of prime, semiprime, strongly prime, irreducible and strongly irreducible intuitionistic fuzzy bi-Γ-ideals in Γ-semigroups. Faisal et al. [6] introduced the notion of (∈,∈∨qk)-fuzzy Γ-ideals of Γ-semigroups. The notion of Q-fuzzy ideals in Γ-semigroups was introduced by Majumder [12]. Lekkoksung [11] gave the concept of Q-fuzzy bi-Γ-ideals of Γ-semigroups. In this paper, we introduce the notion of intuitionistic Q-fuzzy prime (semiprime, strongly prime, irreducible and strongly irreducible) bi-Γ-ideals in Γ-semigroup and we discuss some of the related properties.