Ordered Ehresmann semigroups and categories

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.