Almost Ideals and Fuzzy Almost Ideals of Ordered Semirings

Abstract

An ordered semiring is a system (S,+, ·,≤) consisting of a nonempty set S such that (S,+, ·) is a semiring, (S,≤) is a partially ordered set and for all a, b, c ∈ S, if a ≤ b, then a + c ≤ b + c, c + a ≤ c + b and ac ≤ bc, ca ≤ cb. In this research, we introduce the concepts of almost ordered subsemirings, almost ordered ideals, almost ordered quasi-ideals, almost ordered bi-ideals, almost ordered interior-ideals of ordered semirings and investigate their properties. Moreover, we define fuzzy almost ordered subsemirings, fuzzy almost ordered ideals, fuzzy almost ordered quasi-ideals, fuzzy almost ordered bi-ideals and fuzzy almost ordered interior-ideals of ordered semirings and provide some relationships. Finally, we define tri-quasi ideals and fuzzy tri-quasi ideals of ordered semirings and investigate some properties and relationships between them. In addition, we introduce the notion of almost ordered tri-quasi ideals and fuzzy almost ordered tri-quasi ideals of ordered semirings and give some relationship between them.