On covers of acts over monoids with Condition $(P')$

Year 2020, Volume: 49 Issue: 1, 352 - 361, 06.02.2020

Setareh Irannezhad Ali Madanshekaf

https://doi.org/10.15672/hujms.552216

Cited By: 2

Abstract

In this paper we consider two different definitions of cover, one of them is Enochs' notion of a cover and the other is the one that initiated by Mahmoudi and Renshaw which concerned with the coessential epimorphisms. We show that these definitions are not equivalent in our case and restrict our attention to $(P')$-covers (coessential-covers that satisfy Condition $(P')$). We give a necessary and sufficient condition for a cyclic act to have a $(P')$-cover and a sufficient condition for every act to have a $\mathcal{P'}$-cover (Enochs' $\mathcal{P'}$-cover where $\mathcal{P'}$ is the class of $S$-acts satisfying Condition $(P')$). We also obtain numerous classes of monoids over which indecomposable acts satisfying Condition $(P')$ are locally cyclic.

Keywords

monoids, acts, condition $(P')$, covers

References

• [1] A. Bailey and J. Renshaw, Covers of acts over monoids and pure epimorphisms, Proc. Edinb. Math. Soc. (2), 57(3), 589-617, 2014.
• [2] A. Bailey and J. Renshaw, Covers of acts over monoids II, Semigroup Forum 87(1), 257-274, 2013.
• [3] J. Fountain, Perfect semigroups, Proc. Edinb. Math. Soc. 20(2), 87-93, 1976.
• [4] A. Golchin and H. Mohammadzadeh, On condition $(P')$, Semigroup Forum 86(2), 413-430, 2013.
• [5] John M. Howie, Fundamentals of Semigroup Theory, London Mathematical Society Monographs, Oxford University Press, 1995.
• [6] J. Isbell, Perfect monoids, Semigroup Forum 2, 95-118, 1971.
• [7] M. Kilp, U. Knauer, and A. V. Mikhalev, Monoids, Acts and Categories, with Applications to Wreath Products and Graphs, Walter de Gruyter: Berlin, 2000.
• [8] M. Kilp, Perfect monoids revisited, Semigroup Forum 53(1), 225-229, 1996.
• [9] U. Knauer, Projectivity of acts and Morita equivalence of monoids, Semigroup Forum 3(2), 359-370 1971/1972.
• [10] M. Mahmoudi and J. Renshaw, On covers of cyclic acts over monoids, Semigroup Forum 77(2), 325-338, 2008.
• [11] H. Qiao and L. Wang, On flatness covers of cyclic acts over monoids, Glasg. Math. J., 54(1), 163-167, 2012.
• [12] J. Renshaw, Monoids for which condition (P) acts are projective, Semigroup Forum 61(1), 46-56, 2000.
• [13] J.J. Rotman, An Introduction to Homological Algebra, Academic Press, 1979.