Symmetric groupoids, free categories and E*-unitary inverse monoids

Abstract

Symmetric groupoids which classify the monomorphism contexts of objects in arbitrary categories are studied, along with their connections to symmetric inverse monoids and symmetric inverse algebras. Particular attention is given to symmetric groupoids of objects in free categories and to the inverse algebras induced from them by 0-closure. These graph algebras generalize both the class of polycyclic semigroups and the class of combinatorial ω-semigroups with adjoined zeros. Since all such algebras are E*-unitary, an analogue of McAlister's theory of.E-unitary inverse monoids is developed for a special class of E*-unitary inverse monoids and then illustrated on the class of graph algebras.