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The structure of type A semigroups. (English) Zbl 0605.20057
This article is devoted to the investigation of type A semigroups, introduced by J. B. Fountain [ibid. 28, 285-300 (1977; Zbl 0377.20051)]. Two elements a and b of a semigroup S are said to be \({\mathcal L}^*\)- (\({\mathcal R}^*\)-)related iff a and b are \({\mathcal L}\)- (\({\mathcal R}\)-)related in some oversemigroup of S. A semigroup S is called left (right) abundant if each \({\mathcal L}^*\)- (\({\mathcal R}^*\)-)class contains an idempotent and abundant if it is both left and right abundant. If the idempotents of an abundant semigroup form a semilattice, it is called adequate. If S is adequate, then \(a^*\) \((a^+)\) will denote the unique idempotent in the \({\mathcal L}^*\)- (\({\mathcal R}^*\)-)class of \(a\in S\). An adequate semigroup S is called type A if \(ea=a(ea)^*\) and \(ae=(ae)^+a\) for all elements a in S and all idempotents e in S.
Let \(\sigma\) be a congruence on a type A semigroup such that \(a\sigma\) b if and only if there exists an idempotent e in S with \(ae=be\). A type A semigroup is called proper if \(\sigma\cap {\mathcal L}^*=\iota =\sigma \cap {\mathcal R}^*\). It is shown that every proper type A semigroup is isomorphic to M(T,X,Y) for some admissible triple (T,X,Y). If S is a type A semigroup, then there exists a proper type A semigroup T and a good, idempotent separating homomorphism \(\theta\) : \(T\to S\) onto S. A proper type A semigroup S is fully embeddable in an inverse semigroup if and only if S/\(\sigma\) is embeddable in a group.
Reviewer: P.Normak

20M10 General structure theory for semigroups
20M15 Mappings of semigroups
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