Partial monoid actions and a class of restriction semigroups

We study classes of proper restriction semigroups determined by properties of partial actions underlying them. These properties include strongness, antistrongness, being defined by a homomorphism, being an action etc. Of particular interest is the class determined by homomorphisms, primarily because we observe that its elements, while being close to semidirect products, serve as mediators between general restriction semigroups and semidirect products or $W$-products in an embedding-covering construction. It is remarkable that this class does not have an adequate analogue if specialized to inverse semigroups. $F$-restriction monoids of this class, called ultra $F$-restriction monoids, are determined by homomorphisms from a monoid $T$ to the Munn monoid of a semilattice $Y$. We show that these are precisely the monoids $Y*_mT$ considered by Fountain, Gomes and Gould. We obtain a McAlister-type presentation for the class given by strong dual prehomomorphisms and apply it to construct an embedding of ultra $F$-restriction monoids, for which the base monoid $T$ is free, into $W$-products of semilattices by monoids. Our approach yields new and simpler proofs of two recent embedding-covering results by Szendrei.