Document Type : Research Paper

Author

Department of mathematics, Islamic Azad University, Shahr-e-Qods Branch, Tehran, Iran

Abstract

‎Any notion of purity is normally defined in terms of‎ ‎solvability of some set of equations‎. ‎To study mathematical notions‎, ‎such as injectivity‎, ‎tensor products‎, ‎flatness‎, ‎one needs to have some categorical and‎ ‎algebraic information about the pair (${\mathcal A}$,${\mathcal M}$)‎, ‎for a category $\mathcal A$‎ ‎and a class $\mathcal M$ of monomorphisms in a category $\mathcal A$‎. ‎In this paper we take $\mathcal A$ to be the category {\bf Act-S}‎ ‎of $S$-acts‎, ‎for a semigroup $S$‎, ‎and ${\mathcal M}_{sp}$ to be‎ ‎the class of $C_I^{sp}$-pure monomorphisms and study some‎ ‎categorical and algebraic properties of this class concerning the closure operator $C_I^{sp}$‎.

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