Classification of monoids by Condition $(PWP_{ssc})$

Pouyan Khamechi,Leila Nouri,Hossein Mohammadzadeh Saany,Zahedan Baluchestan

doi:10.29252/cgasa.12.1.175

Abstract

Condition $(PWP)$ which was introduced in (Laan, V., {it Pullbacks and flatness properties of acts I}, Commun. Algebra, 29(2) (2001), 829-850), is related to flatness concept of acts over monoids. Golchin and Mohammadzadeh in ({it On Condition $(PWP_E)$}, Southeast Asian Bull. Math., 33 (2009), 245-256) introduced Condition $(PWP_E)$, such that Condition $(PWP)$ implies it, that is, Condition $(PWP_E)$ is a generalization of Condition $(PWP)$. In this paper we introduce Condition $(PWP_{ssc})$, which is much easier to check than Conditions $(PWP)$ and $(PWP_E)$ and does not imply them. Also principally weakly flat is a generalization of this condition. At first, general properties of Condition $(PWP_{ssc})$ will be given. Finally a classification of monoids will be given for which all (cyclic, monocyclic) acts satisfy Condition $(PWP_{ssc})$ and also a classification of monoids $S$ will be given for which all right $S$-acts satisfying some other flatness properties have Condition $(PWP_{ssc})$.

Highlights

For a monoid S, with 1 as its identity, a set A is called a right S-act, usually denoted by AS, if S acts on AP
The study of flatness properties of S-acts in general began in the early 1970s and a comprehensive survey of this research is found in [14]
In this paper we introduce Condition (P W Pssc) and compare it with principally weakly flat, Condition (P W P ) and Condition (P W PE)

Summary

Introduction

(2) For left PSF monoid S, A is principally weakly flat if and only if A satisfies Condition (P W Pssc). Since S is left PSF, principally weakly flat is equivalent to Condition (P W Pssc), by Theorem 2.8. Since every regular monoid is left PP and so is left PSF, principally weakly flat is equivalent to Condition (P W Pssc), by part (2) of Theorem 2.8.

Results

Conclusion