Document Type : Research Paper


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


‎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 $C_I^{sp}$‎. 


Main Subjects

[1] B. Banaschewski, Injectivity and essential extensions in equational classes of algebras, Queen’s Papers in Pure. Appl. Math. 25 (1970), 131-147.
[2] H. Barzegar, Sequentially Complete S-acts and Baer type criteria over semigroups, European J. pure. Appl. Math. 6 (2)(2013), 211-221.
[3] H. Barzegar, Strongly s-dense monomorphism, J. Hyperstructures. 1 (1) (2012), 14-26.
[4] H. Barzegar, Essentiality in the Category of S-acts, European J. pure. Appl. Math. 9 (1) (2016), 19-26.
[5] P. Berthiaume, The injective envelope of S-Sets, Canad. Math. Bull. 10 (2) (1967), 261-273.
[6] D. Dikranjan, W. Tholen, Categorical Structure of Closure Operators, with Applications to Topology, Algebra, and Discrete Mathematics, Mathematics and Its Applications, Kluwer Academic Publication, 1995.
[7] M. M. Ebrahimi, M. Mahmoudi, Purity and equational compactness of projection algebras, Appl. Categ. Struc. 9 (2001), 381-394.
[8] M. Ghorbani, H. Rahimi, On Biprojectivity and Biflatness of Banach algebras, Inter. J. Appl. Math. Statistics. 58 (3) (2019), 82-89.
[9] M. Ghorbani, H. Rahimi, Biprojectivty of Banach algebras modulo an ideal, J. New Researches in Mathematics. 5 (18) (2019), 21-30.
[10] V. Gould, The characterisation of monoids by properties of their S-systems, Semigroup Forum. 32 (3) (1985), 251-265.
[11] M. Kilp, U. Knauer, A. Mikhalev, Monoids, Acts and Categories, Walter de Gruyter, Berlin, New York, 2000.
[12] P. Normak, Purity in the category of M-sets, Semigroup Forum. 20 (2) (1980), 157-170.