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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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