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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 Fk FF and Fk θ ◦k FF, where θ (x) = 0 for all xG.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

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