Symmetric groupoids, free categories andE*-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 u>-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 £*-unitary inverse monoids and then illustrated on the class of graph algebras.Symmetric groupoids, free categories and inverse monoids 961 w-semigroups with adjoined zeros. Their algebraic properties are studied in terms of the generating graphs of their ambient categories throughout § §4 and 5.Section 5 is concerned with £"*-unitary inverse monoids, of which graph algebras are instances. Just as i?-unitary inverse monoids arise as the symmetric inverse monoids of objects in cancellative categories having finite intersections, so E*unitary inverse monoids arise in cancellative categories having conditional intersections which, like free categories, must be augmented by initial objects to obtain finite intersections. Equivalently, i?*-unitary inverse monoids arise as O-completions of what might be termed 'is-unitary' groupoids, but are perhaps more appropriately termed 'fixed point free' groupoids. Of special interest is a class of i?*-unitary inverse monoids which possesses a modified McAlister theory, or better, a hybrid McAlister-Rees theory. While details are perhaps best reserved for later, suffice it to say that this theory is amply illustrated in the case of graph algebras. Contexts and symmetric groupoidsDEFINITION 2.1. A groupoid is any small category G of isomorphisms. Given any morphism fi of G, its domain identity is d(n) = \i~ 1 fi, and its co-domain identity (or range identity) is r(/i) = fifJ.' 1 -An ordered groupoid is firstly any pair (G, ^) where G is a groupoid and î s a partial ordering of the isomorphisms compatible with both composition and inversion. That is, given \i ^ v and a ^ r in G, then [ia ^ VT. provided fia and VT exist in G, and also /i^1 ^ v~l. This ordering induces an ordering of the objects of G given by:and r(/i) ^ r(v) and hence both dom(/z) ^ dom(^) and cod(/i) ^ cod(V), where dom and cod denote the domain and co-domain of the respective morphisms.Secondly, given any morphism v and any object A in an ordered groupoid (G, ^), the following equivalent assertions are assumed to hold:

Partialization of Categories and Inverse Braid-Permutation Monoids

“…In particular, if C 9 has one object and all morphisms in C 9 are monomorphisms, then Condition I describes exactly the situation, dual to the one discussed in [20, 1.2]. This approach was further developed, in particular, in [21,22].…”

“…We also formulate some consequences for the structural theory of the semigroups of endomorphisms which arise from partialization. Again there are multiple precedents in the literature, particularly in the case of inverse semigroups; one may mention [21,22] for a mature development. We present a full development in a form suited to our present purposes.…”