A new type of the hybrid algebra between Abelian groups and UP (BCC)-algebras: UP (BCC)-modules

Abstract

The goal of this study is to introduce the concept of a new type of the hybrid algebra between Abelian groups and UP (BCC)-algebras: UP (BCC)-modules. We introduce the concept of fuzzy UP (BCC)-submodules of UP (BCC)-modules and provide properties and find the necessary and sufficient conditions for this concept. We define fuzzy sets in UP (BCC)-modules of many forms, supplying their properties and their relation to fuzzy UP (BCC)-submodules. We also define and study the fuzzy UP (BCC)-submodule generated by a set of fuzzy sets in UP (BCC)-modules, as well as provide for their properties and their relation to fuzzy UP (BCC)-submodules. Finally, we apply the concept of fuzzy UP (BCC)-ideals of UP (BCC)-algebras while providing properties and find the results of the composition and the product between fuzzy UP (BCC)-ideals and fuzzy UP (BCC)-submodules.