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Classification of monoids by condition \((PWP_{ssc})\). (English) Zbl 1446.20081
Summary: Condition \((PWP)\) which was introduced in [V. Laan, Commun. Algebra 29, No. 2, 829–850 (2001; Zbl 0987.20047)], is related to flatness concept of acts over monoids. A. Golchin and H. Mohammadzadeh [Southeast Asian Bull. Math. 33, No. 2, 245–256 (2009; Zbl 1203.20062)] 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 \((PWP_{ssc})\).
MSC:
20M50 Connections of semigroups with homological algebra and category theory
20M30 Representation of semigroups; actions of semigroups on sets
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