| 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 (xin 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 (fin S_G) such that the map (xmapsto 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. |
