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On a wreath product embedding and idempotent pure congruences on inverse semigroups. (English) Zbl 0769.20027

The related notions of a \(\lambda\)-semidirect product and of a \(\lambda\)- wreath product of two inverse semigroups are introduced. It is proved that if \(S\) is an inverse semigroup, \(\rho\) a congruence on \(S\) such that for each \(\rho\)-class \(s\rho\), the set \(\{x^{-1}x:x \in s\rho\}\) contains a greatest element with respect to the natural partial order, then \(S\) can be embedded in the \(\lambda\)-wreath product of \(\text{Ker }\rho\) and \(S/\rho\), where \(\text{Ker }\rho = \{s \in S: s\rho e\), for some \(e = e^ 2\}\).
It is proved that if \(\rho\) is an idempotent pure congruence on \(S\) then \(S\) has a representation as a certain subsemigroup of a \(\lambda\)- semidirect product of a semilattice and \(S/\rho\). From the results given the McAlister \(P\)-theorem can be recovered along with O’Carroll’s result that any \(E\)-reflexive inverse semigroup is embeddable in a strong semilattice of inverse semigroups, each of which is a semidirect product of a semilattice and a group.

MSC:

20M18 Inverse semigroups
20M10 General structure theory for semigroups
20M15 Mappings of semigroups
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References:

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