A left topological monoid associated to a topological groupoid

Abstract

This paper presents a fanctor $S$ from the category of groupoids to the category of semigroups. Indeed, a monoid $S_G$ with a right zero element is related to a topological groupoid $G$. The monoid $S_G$ is a subset of $C(G,G)$, the set of all continuous functions from $G$ to $G$, and with the compact- open topology inherited from C(G,G) is a left topological monoid. The group of units of $S_G$, which is denoted by $H(1)$, is isomorphic to a subgroup of the group of all bijection map from $G$ to $G$ under composition of functions. Moreover, it is proved that $H(1)$ is embedded in the group of all invertible linear operators on $C(G)$, the set of all complex continuous function on $G$.