The Szendrei expansion of restriction semigroupoids

Abstract We introduce the concept of a restriction semigroupoid script upper S S $\mathcal{S}$ , 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 ) $Sz(\mathcal{S})$ of script upper S S $\mathcal{S}$ and establish that each premorphism between two restriction semigroupoids script upper S S $\mathcal{S}$ and script upper T T $\mathcal{T}$ is uniquely factorized by a morphism between the Szendrei expansion upper S z left parenthesis script upper S right parenthesis S z ( S ) $Sz(\mathcal{S})$ and script upper T T $\mathcal{T}$ .