Ordered Ehresmann semigroups and categories

TL;DR

This paper introduces ordered Ehresmann semigroups, explores their properties, and characterizes restriction semigroups within this framework, culminating in an ESN-style theorem for these structures.

Contribution

It defines ordered Ehresmann semigroups, analyzes their properties, and provides a characterization of restriction semigroups within this class, including an ESN-style theorem.

Findings

Restriction semigroups are characterized within ordered Ehresmann semigroups.

Ordered Ehresmann semigroups generalize biunary semigroups with domain and range operations.

An ESN-style theorem is established for these semigroups.

Abstract

Ehresmann semigroups may be viewed as biunary semigroups equipped with domain and range operations satisfying some equational laws. Motivated by some of the main examples, we here define ordered Ehresmann semigroups, and consider their basic properties as well as special cases in which the order is algebraically definable. In particular, one and two-sided restriction semigroups equipped with their natural orders are characterised within the class of ordered Ehresmann semigroups. The main result is an ESN-style theorem for ordered Ehresmann semigroups with particular reference to the special cases.