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 $$S_{Γ}$$-acts via a congruence relation, the derivation of universal factorization properties for morphisms and the Local-Global Principle for $$S_{Γ}$$-act structures. The concept of $$R_{Γ}$$-multiplication $$S_{Γ}$$-acts is then introduced as a generalization of multiplication $$S_{Γ}$$-acts with a complete characterization of their basic properties. The relationship between multiplication $$S_{Γ}$$-acts and $$R_{Γ}$$-multiplication $$S_{Γ}$$-The act is examined. In particular, we prove that every multiplication $$S_{Γ}$$-act is an $$R_{Γ}$$-multiplication $$S_{Γ}$$-act, but the converse does not always hold. Using the concept of idempotent $$S_{Γ}$$-subacts, we show that every $$S_{Γ}$$-subact is an $$R_{Γ}$$-multiplication $$S_{Γ}$$-acts. Furthermore, additional concepts such as $$R_{Γ}$$-cyclic and $$R_{Γ}$$-finite $$S_{Γ}$$-Acts are introduced and some of their properties are investigated. Finally, the notion of an $$R_{Γ}$$-finite $$S_{Γ}$$-Subact is used to study the relationship between $$R_{Γ}$$-cyclic and $$R_{Γ}$$-multiplication $$S_{Γ}$$-acts, and several important properties are discussed.

Keywords: Localization of $$S_{Γ}$$-acts, Universal Property of Localization, Γ-semigroup, multiplication $$S_{Γ}$$-acts, $$R_{Γ}$$-cyclic $$S_{Γ}$$-subact, $$R_{Γ}$$-finite $$S_{Γ}$$-subact, $$R_{Γ}$$-multiplication $$S_{Γ}$$-acts.

Pages: 777-786