A group of continuous self-maps on a topological groupoid

Abstract

The group of bisections of groupoids plays an important role in the study of Lie groupoids. In this paper another construction is introduced. Indeed, for a topological groupoid G, the set of all continuous self-maps f on G such that (x, f(x)) is a composable pair for every $x\in G$, is denoted by $S_G$. We show that $S_G$ by a natural binary operation is a monoid. $S_G(\alpha )$, the group of units in $S_G$ precisely consists of those $f\in S_G$ such that the map $x\mapsto xf(x)$ is a bijection on G. Similar to the group of bisections, $S_G(\alpha )$ acts on G from the right and on the space of continuous self-maps on G from the left. It is proved that $S_G(\alpha )$ with the compact- open topology inherited from C(G, G) is a left topological group. For a compact Hausdorff groupoid G it is proved that the group of bisections of $G^2$ is isomorphic to the group $S_G(\alpha )$ and the group of transitive bisections of G, $Bis_T(G)$, is embedded in $S_G(\alpha )$, where $G^2$ is the groupoid of all composable pairs.