A P-theorem for Inverse Semigroupoids through Ordered Globalizations

TL;DR

This paper proves a P-theorem for inverse semigroupoids, showing they can be represented as semidirect products from ordered partial actions, extending classical results to a multi-object setting.

Contribution

It introduces a multi-object P-theorem for inverse semigroupoids, connecting ordered partial actions of groupoids with a generalized McAlister triple framework.

Findings

Every ordered partial action admits a globalization.

Established a connection between groupoid actions and McAlister triples.

Proved that E-unitary inverse semigroupoids are isomorphic to certain semidirect products.

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.