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Proper left type-\(A\) monoids revisited. (English) Zbl 0802.20051
The relation \({\mathcal R}^*\) on a semigroup \(S\) is defined by \(a{\mathcal R}^* b\) if \(a {\mathcal R}b\) in some oversemigroup of \(S\). \(S\) is an \(E\)- semigroup if its set \(E(S)\) of idempotents is a subsemilattice of \(S\). An \(E\)-semigroup is left adequate if every \({\mathcal R}^*\)-class contains an idempotent. This idempotent is unique and denoted for \(a \in S\) by \(a^ +\). A left adequate semigroup is left type-\(A\) if \(ae = (ae)^ + a\) for each \(a \in S\) and \(e \in E(S)\). On a left type-\(A\) semigroup there is a minimum right cancellative congruence \(\sigma\). A left type-\(A\) semigroup is called proper if \(\sigma \cap {\mathcal R}^* = \iota\). Several characterizations of proper left type-\(A\) monoids and \(E\)-unitary left type-\(A\) monoids are presented. For instance, every \(E\)-unitary (resp. proper) left type \(A\) monoid is isomorphic to a monoid \(C_ u\), where \(u\) is an object of a (proper) idempotent left type-\(A\) category on which a right cancellative monoid \(T\) acts, and \(C_ u = \{(t,p)\mid t\in T,\;p\in \text{Mor}(ut,u)\}\) and multiplication is defined by \((t,p)(h,q) = (th,ph + q)\).
Reviewer: J.Henno (Tallinn)

MSC:
20M10 General structure theory for semigroups
20M50 Connections of semigroups with homological algebra and category theory
20M15 Mappings of semigroups
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