Almost Ideals and Fuzzy Almost Ideals in Algebraic Structures

Abstract

A semigroup is an ordered pair (S, .), where S is a nonempty set and . is an associative binary operation. The semigroup (S, .) with a partial order <= is an ordered semigroup if x <= y, then x.z <= y.z and z.x <= z.y for all x, y, z in S. A semihypergroup (H, *) can be defined in a similar way to the semigroup, but the operation * of the semihypergroup is a function from H x H into P^*(H), where P^*(H) is a set of all nonempty subsets of H. In this thesis, we define almost (m, n)-ideals and fuzzy almost (m, n)-ideals in semigroups and study some of their properties. In addition, we define ordered almost ideals, ordered almost bi-ideals, ordered almost quasi-ideals, fuzzy ordered almost ideals, fuzzy ordered almost bi-ideals and fuzzy ordered almost quasi-ideals in ordered semigroups and we give the relations of them. Moreover, we define almost hyperideals, almost bi-hyperideals and almost quasi-hyperideals in semihypergroups, and give some interesting properties and relations of them.