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Proper left type-\(A\) covers. (English) Zbl 0812.20043
A monoid where all principal left ideals are projective is called left abundant (also called left PP). A left abundant monoid is left type-\(A\) if the set \(E(S)\) of idempotents of \(S\) is a commutative submonoid of \(S\) and \(eS \cap aS = eaS\) for all \(e \in E(S)\) and \(a \in S\). A relation \({\mathcal R}^*\) is defined as follows: \((a,b) \in {\mathcal R}^* \Leftrightarrow [(\forall s,t\in S)\) \(sa = ta \Leftrightarrow sb = tb]\). A left type-\(A\) monoid is called proper if \(a{\mathcal R}^* b\), \(a \neq b\), imply \(ea \neq eb\) for al \(e \in E(S)\). The authors study left type-\(A\) monoids by means of left actions on categories. Using techniques developed in the article left type-\(A\) monoids are characterized in terms of \(M\)-systems. It is also proved that every left type-\(A\) monoid \(M\) has a proper left type-\(A^ +\) cover (i.e. there exist a proper type-\(A\) monoid \(P\) and an idempotent separating surjective homomorphism \(\Theta:P \to M\) such that \(a^ + \Theta = (a \Theta)^ +\)).
Reviewer: P.Normak (Tallinn)
20M10 General structure theory for semigroups
20M50 Connections of semigroups with homological algebra and category theory
18B40 Groupoids, semigroupoids, semigroups, groups (viewed as categories)
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