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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)

20M10 General structure theory for semigroups
20M50 Connections of semigroups with homological algebra and category theory
20M15 Mappings of semigroups
Full Text: DOI
[1] DOI: 10.2307/1997032 · Zbl 0297.20072
[2] DOI: 10.2307/1996831 · Zbl 0297.20071
[3] DOI: 10.1016/0021-8693(87)90046-9 · Zbl 0625.20043
[4] DOI: 10.1016/0021-8693(76)90023-5 · Zbl 0343.20037
[5] DOI: 10.1016/0022-4049(92)90066-O · Zbl 0774.20047
[6] DOI: 10.1017/S0013091500016230 · Zbl 0414.20048
[7] DOI: 10.1093/qmath/28.3.285 · Zbl 0377.20051
[8] Margolis, Proc. 1984 Marquette Conf. on Semigroups pp 85– (1985)
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