A New Generalization of Fuzzy Bi-ideals in Semigroups and Its Applications in Fuzzy Finite State Machines
Saleem Abdullah, Kostaq Hila and Muhammad Aslam
In this paper, we introduce a generalization of anti fuzzy bi-ideals of semigroups of type (α, β)*-fuzzy bi-ideals, where α, β ∈ {<, γk, < ∨γk, < ∧γk} with α ≠ < ∧γk. We prove some basic results of (α, β)*-fuzzy bi-ideals of semigroups. Moreover, we give characterizations of regular semigroups by the properties of (α, β)*-fuzzy bi-ideals. We also prove that a fuzzy set F in a semigroup G is an (<,< ∨γk)*-fuzzy bi-ideal of G if and only if F ◦k F ⊇ F and F ◦k θ ◦k F ⊇ F, where θ (x) = 0 for all x ∈ G.We define an anti fuzzy finite state machine and an (<,< ∨γk)*-fuzzy transformation semigroup. We describe some applications of the given concept to an (<,< ∨γk)*-fuzzy transformation semigroup and an anti fuzzy finite state machine.
Keywords: Semigroup, fuzzy set, bi-ideals, (α, β)*-fuzzy bi-ideals, (<, < ∨γk)*-fuzzy bi-ideal