The Szendrei expansion of restriction semigroupoids

We introduce the concept of a restriction semigroupoid script upper S S S , which unifies the notion of restriction semigroups and restriction categories within a single structure. We prove a representation theorem, showing that every restriction semigroupoid can be embedded into a certain category of partial maps. Furthermore, we construct the Szendrei expansion upper S z left parenthesis script upper S right parenthesis S z ( S ) Sz(S) of script upper S S S and establish that each premorphism between two restriction semigroupoids script upper S S S and script upper T T T is uniquely factorized by a morphism between the Szendrei expansion upper S z left parenthesis script upper S right parenthesis S z ( S ) Sz(S) and script upper T T T .