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Extensions and covers for semigroups whose idempotents form a left regular band. (English) Zbl 1237.20055
The generalized prefix expansion $$S^{Pr}$$ of an $$\mathcal R$$-unipotent semigroup was proved to be an idempotent pure extension of $$S$$ through the second projection $$\eta_S$$ which is injective on $$\mathcal L$$-related idempotents [see M. Branco, G. M. S. Gomes, Acta Math. Hung. 123, No. 1-2, 11-26 (2009; Zbl 1188.20070)]. However, not all idempotent pure extensions of an $$\mathcal R$$-unipotent semigroup are of this kind. The first task of this article is to describe such special extensions by means of a variation of a semidirect product of a left regular band by another $$\mathcal R$$-unipotent semigroup; the variation we are talking about is Billhardt’s notion of $$\lambda$$-semidirect product, introduced by B. Billhardt [in Semigroup Forum 45, No. 1, 45-54 (1992; Zbl 0769.20027)] for inverse semigroups.
Let $$S$$ be a semigroup and $$E\subseteq E(S)$$, where $$E(S)$$ denotes the set of all its idempotents. The relation $$\widetilde{\mathcal R}_E$$ on $$S$$ is defined by the rule that for all $$a,b\in S$$, $$a\widetilde{\mathcal R}_Eb$$ if and only if $$\{e\in E:ea=a\}=\{e\in E:eb=b\}$$.
The authors are interested in semigroups $$S$$ in which every $$\widetilde{\mathcal R}_E$$-class contains a unique idempotent of $$E$$. In this case, it is denoted by $$a^+$$ the idempotent of $$E$$ in the $$\widetilde{\mathcal R}_E$$-class of $$a$$, so that $$a\mapsto a^+$$ is a unary operation on $$S$$ and $$S$$ may be regarded as an algebra of type $$(2,1)$$. It is easy to see that $$S$$ satisfies the identities $$x^+x=x$$, $$(x^+)^+=x^+$$, $$(x^+y^+)^+=x^+y^+$$, $$x^+y^+x^+=x^+y^+$$, $$x^+x^+=x^+$$, $$x^+(xy)^+=(xy)^+$$. Conversely, let $$S$$ be an algebra of type $$(2,1)$$ where the binary operation is associative and the unary operation is written as $$^+$$, satisfying the above identities. Put $$E=\{a^+:a\in S\}$$. Then $$E$$ is a subband and for any $$a\in S$$, the $$\widetilde{\mathcal R}_E$$-class of $$a$$ contains a unique idempotent $$a^+$$. If the above conditions hold, then $$E$$ is a left regular band. The algebras of type $$(2,1)$$ satisfying the above identities are called generalized left restriction semigroups and $$E$$ is called the distinguished band. The generalized left restriction semigroups satisfying the ample identity $$(xy^+)^+x=xy^+$$ and the left congruence identity $$(xy^+)^+=(xy)^+$$ are called glrac semigroups.
Let $$S$$ be a glrac semigroup which is a proper $$\mathcal L_F$$-extension of a glrac semigroup $$T$$. Then $$S$$ is embeddable into a $$\lambda$$-semidirect product of a left regular band by $$T$$. After that are formulated and proved covering theorems for generalized left restriction semigroups. In conclusion the authors give a structure theorem for proper left restriction semigroups which may be regarded as an analogue of the McAlister $$P$$-theorem [see D. B. McAlister, Trans. Am. Math. Soc. 196, 351-370 (1974; Zbl 0297.20072)].

##### MSC:
 20M10 General structure theory for semigroups
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##### References:
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