## 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)

### MSC:

 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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