A closure operator versus purity

Subject Areas : Operator theory

M.  Ghorbani ¹ (Department of mathematics, Islamic Azad University, Shahr-e-Qods Branch, Tehran, Iran)

Received: 2021-06-09 Accepted : 2021-08-11 Published : 2021-09-01

Keywords: semigroup, purity, closure operator, $S$-act,

Abstract :

‎Any notion of purity is normally defined in terms of‎‎solvability of some set of equations‎.‎To study mathematical notions‎, ‎such as injectivity‎,‎tensor products‎, ‎flatness‎, ‎one needs to have some categorical and‎‎algebraic 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 be‎‎the class of $C_I^{sp}$-pure monomorphisms and study some‎‎categorical and algebraic properties of this class concerning the closure operator $C_I^{sp}$‎.

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