FUZZY IDEALS IN VECTOR VALUED SEMIGROUPS

Abstract

In this paper we introduce the concepts of fuzzy subset of vector valued groupoid (semigroup), as well as fuzzy subgroupoid, fuzzy subsemigroup, fuzzy \(i\)-ideal (ideal) and bi-ideal of vector valued groupoid (semigroup), investigate their properties and present suitable examples. Prime and fuzzy prime, semiprime and fuzzy semiprime subsets of vector valued groupoids are defined and their properties are investigated. We characterize the Green's relations \(\mathcal{J}_i\) on a vector valued semigroup \(\mathbf{S}\) in terms of fuzzy subsets. Green's relations \(\mathcal{J}_i^F\) on \(\mathbf{S}\) are suitably defined and it is shown that they coincide with the relation \(\mathcal{J}_i\) on \(\mathbf{S}\).