Inverse semigroups with certain universal properties

Abstract

Free groups and free inverse semigroups are characterized by certain properties which are usually called universal. Category-theoretically they arise from left adjoints of forgetful functors which assign to each structure its underlying set. The purpose of this note is to give a construction of a wealth of inverse semigroups with certain universal properties which subsumes free groups and free inverse semigroups and incidentally elucidates some well-known constructions of free inverse semigroups (Scheiblich (1972)). Categorytheoretical terminology will be freely taken from (Mac Lane (1971)).