On (α, β)-Intuitionistic Fuzzy Ideals of Hemirings

Abstract

Dedekind introduce the modern definition of the ideal of a ring in 1894 and observed that the family Id(R) of all the ideals of a ring R obeyed most of the rules that the ring(R, +, ⋅) did, but (Id(R), +, ⋅) was not a ring. In 1934, Vandiver [18] studied an algebraic system, which consisted of a non-empty set S with two binary operations "+" and "⋅" such that S was a semigroup under both the operations and (S, +, ⋅) satisfied both the distributive laws but did not satisfy the cancellation law of addition. Vandiver named this system a `semiring'. Semirings are common generalization of rings and distributive lattices. A hemiring is a semiring in which "+" is commutative and it has an absorbing element. Semirings (hemirings) appear in a natural manner in some applications to the theory of automata, formal languages, optimization theory and other branches of applied mathematics (see for example [10]). Zadeh introduced the concept of fuzzy set in his definitive paper [20] of 1965. Many authors used this concept to generalize basic notions of algebra. In 1971, Rosenfeld laid the foundations of fuzzy algebra. He introduced the notions of fuzzy subgroup of a group. Ahsan et al. [3] initiated the study of fuzzy semirings. Murali defined the concept of belongingness of a fuzzy point to fuzzy subset under a natural equivalence on fuzzy subset and Pu and Liu introduced the concept of quasic-oincident of a fuzzy point with a fuzzy set. Bhakat and Das used these ideas and defined ( q ∨ εε , )-fuzzy subgroup of a group which is a generalization of Rosenfeld’s fuzzy subgroups. Many researchers used these ideas to define ( βα , )-fuzzy substructures of algebraic structures. Generalizing the concept of the quasi-coincident of a fuzzy point with a fuzzy subset, Jun [13] defined ( k q∨ εε , )-fuzzy subalgebra in BCK/BCI-algebras. Jun et al. in [14] defined ( k q∨ εε , )fuzzy ideals of hemirings. On the other hand Atanassov [5] introduced the notion of intuitionistic fuzzy set which is a generalization of fuzzy set. Intuitionistic fuzzy hemirings are studied by Dudek. Coker and Demirici [8] introduced the notion of intuitionistic fuzzy point with an intuitionistic fuzzy set. Jun introduced the notion of ( ψφ , )-intuitionistic fuzzy subgroup of a group where { } qqq ∧∨ εε εε ψφ ,,,, and q ∧≠ εφ . Abdullah et al. in [1] introduced ( βα , )-intuitionistic fuzzy ideals of hemirings. In this dissertation, generalizing the concept of quasi-coincident of an intuitionistic fuzzy point with an intuitionistic fuzzy set, we define ( k q∨ εε , )-intuitionistic fuzzy ideals of hemirings and characterize different classes of hemirings by the properties of these ideals. This dissertation consists of three chapters. The first chapter is devoted to fundamental concepts of hemirings and fuzzy hemirings. In the second chapter, we introduce the concept of a ( k q∨ εε , )intuitionistic fuzzy sub-hemiring and ( k q∨ εε , )-intuitionistic fuzzy ideals in hemirings and check out their properties with their level subsets. We have also characterize hemirings for which each ( k q∨ εε , )-intuitionistic fuzzy ideal is idempotent. In the third chapter, we have defined the ( k q∨ εε , )-intuitionistic fuzzy k-ideals in hemirings. We have also introduce the notion of ( k q∨ εε , )-intuitionistic fuzzy k-quasi-ideals and ( k q∨ εε , )-intuitionistic fuzzy k-bi-ideals in hemirings and characterize regular, k-regular, and k-intra-regular hemirings with the help of these ideals.