On R-multiplication Gamma Acts

Source abstract

This article introduces and studies Γ-multiplicatively closed subsets R of a Γ-monoid S, developing their foundational properties through definitions and characterization theorems and essential structural results. The work extends classical localization and ideal theory to Γ-semigroups through three principal advances: the localization of 𝑆Г -acts via a congruence relation, the derivation of universal factorization properties for morphisms and the Local-Global Principle for 𝑆Г -act structures. The concept of 𝑅Г -multiplication 𝑆Г -acts is then introduced as a generalization of multiplication 𝑆Г -acts with a complete characterization of their basic properties. The relationship between multiplication 𝑆Г -acts and 𝑅Г -multiplication 𝑆Г -The act is examined. In particular, we prove that every multiplication 𝑆Г -act is an 𝑅Г -multiplication 𝑆Г -act, but the converse does not always hold. Using the concept of idempotent 𝑆Г -subacts, we show that every 𝑆Г -subact is an 𝑅Г -multiplication 𝑆Г -acts. Furthermore, additional concepts such as 𝑅Г -cyclic and 𝑅Г -finite 𝑆Г -Acts are introduced and some of their properties are investigated. Finally, the notion of an 𝑅Г -finite 𝑆Г -Subact is used to study the relationship between 𝑅Г -cyclic and 𝑅Г -multiplication 𝑆Г -acts, and several important properties are discussed.