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Proper restriction semigroups – semidirect products and $$W$$-products. (English) Zbl 1295.20064
Left restriction semigroups are unary semigroups isomorphic to unary subsemigroups of partial transformation semigroups, where the unary operation is $$x\to x^+=I_{\mathrm{dom}\;x}$$. The semilattice of projections of a left restriction semigroup $$S$$ is $$E=\{x^+,\;x\in S\}$$; $$\sigma_S$$ is the least congruence identifying all the elements of $$E$$. A left restriction semigroup is proper if $$a^+=b^+$$, $$a\sigma_Sb$$ yields $$a=b$$, and left ample if $$xz=yz$$ yields $$xz^+=yz^+$$.
It is known that any proper left ample semigroup embeds into a so-called $$W$$-product, which is a subsemigroup of a reverse semidirect product of a semilattice $$\mathcal Y$$ by a monoid $$T$$, where the action of $$T$$ on $$\mathcal Y$$ is injective with images of the action being order ideals of $$\mathcal Y$$. Here necessary and sufficient conditions are given for a proper left restriction semigroup to be embeddable into a $$W$$-product.

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