SP-Fuzzy Soft Ideals in Semigroups

Abstract

In this paper, the definition of a new concept which is a member of the class $\tilde{(U,P)}$ and which is referred to as $UP$-fuzzy soft subset of a soft set on the class $(U,P)$ is introduced, where $\tilde{(U,P)}$  denotes the fuzzy soft class and $(U,P)$ denotes the soft class with the universal set $U$ and the set of parameters $P$. We give the definitions of the complement and $\alpha$-level soft set of a $UP$-fuzzy soft subset of a soft set. It is demonstrated that $UP$-fuzzy soft subsets provide De Morgan rules for restricted union and restricted intersection. Furthermore, considering a semigroup $S$ as an universal set, this paper presents some new algebraic notions which are called $SP$-fuzzy soft subsemigroup and $SP$-fuzzy soft left (right, bi-, quasi, interior) ideal of a soft semigroup. We examine some basic properties such as restricted union, extended union, restricted intersection, extended intersection and product of the families of $SP$-fuzzy soft subsemigroups and $SP$-fuzzy soft left (right, bi-, quasi, interior) ideals. It is obtained that the restricted intersection of the family of $SP$-fuzzy soft subsemigroups is a $SP$-fuzzy soft subsemigroup of the restricted intersection of the family of soft sets. Moreover it is indicated that an $\alpha$-level soft set of a $SP$-fuzzy soft subset is a soft subsemigroup for all $\alpha \in [0,1]$ if and only if the SP-fuzzy soft subset is a SP-fuzzy soft subsemigroup.

Keywords

• (Fuzzy) soft set,UP-fuzzy soft subset,SP-fuzzy soft subsemigroup

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