A P-theorem for Inverse Semigroupoids through Ordered Globalizations
Felipe Augusto Tasca, Paulinho Demeneghi, Víctor Marín, Willian Goulart Gomes Velasco
TL;DR
This work extends the classical $P$-theorem from groups to the multi-object setting of inverse semigroupoids by developing a globalization theory for ordered partial actions on posets. It introduces the multi-object analogues of McAlister triples and semilatticeoids, and shows that every $E$-unitary inverse semigroupoid $\mathcal{S}$ arises as a semidirect product $\mathcal{S}/\sigma \ltimes_{\alpha} E(\mathcal{S})$, where $\mathcal{S}/\sigma$ is the maximal groupoid image and $\alpha$ is the Munn-type action on $E(\mathcal{S})$. The construction hinges on the existence and universality of ordered globalizations, and it connects ordered partial groupoid actions to McAlister triples via the TMP, yielding a robust multi-object analogue of the $P$-theorem. The results unify and extend structural insights for $E$-unitary inverse semigroups, with potential applications to multi-object symmetry and partial action theories in higher-categorical settings.
Abstract
We prove that every ordered partial action of an inverse semigroupoid on a partially ordered set admits a globalization. This result is used to establish a connection between ordered partial actions of groupoids and a multi-object analogue of McAlister triples. As a consequence, we obtain a multi-object version of the P-theorem: every E-unitary inverse semigroupoid is isomorphic to a semidirect product arising from an ordered partial action of a groupoid on a multi-object version of a semilattice.