We study a class of morphisms between two-sided restriction semigroups which we call {\it proper morphisms}. This is a generalization of the notion of an idempotent pure homomorphism of inverse semigroups. With each proper morphism $\psi:S\to T$ we associate two partial actions $\widetilde\psi$ and $\widehat\psi$ of $T$ on $P(S)$, such that $S\cong P(S)\rtimes T$ with respect to each of the two underlying partial actions. We thus obtain a generalization of the Cornock's and Gould's result on the description of proper restriction semigroups, as well as the partial action version of the O'Carroll's result on idempotent pure extensions of inverse semigroups.

This is a joint work with Mikhailo Dokuchaev (Universidade de Sao Paulo) and Ganna Kudryavtseva (University of Ljubljana).