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 Gsuch that (x, f(x)) is a composable pair for every ????∈????x∈G, is denoted by ????????SG. We show that ????????SG by a natural binary operation is a monoid. ????????(????)SG(α), the group of units in ????????SG precisely consists of those ????∈????????f∈SG such that the map ????↦????????(????)x↦xf(x) is a bijection on G. Similar to the group of bisections, ????????(????)SG(α) acts on G from the right and on the space of continuous self-maps on G from the left. It is proved that ????????(????)SG(α) 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 ????2G2 is isomorphic to the group ????????(????)SG(α) and the group of transitive bisections of G, ????????????????(????)BisT(G), is embedded in ????????(????)SG(α), where ????2G2 is the groupoid of all composable pairs.