Partial actions and proper extensions of two-sided restriction semigroups

We prove a structure result on proper extensions of two-sided restriction semigroups in terms partial actions, generalizing respective results for monoids and inverse upgrading the latter. introduce study several classes actions that generalize semigroups. establish an adjunction between category P(S) semigroup S A(S) subject to certain conditions going back work O'Carroll. In A(S), we specify two isomorphic subcategories, one being reflective other coreflective, each which is equivalent P(S).