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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

##### MSC:
 20M10 General structure theory for semigroups 20M15 Mappings of semigroups
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