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)].