A note on topological semigroup-groupoid (original) (raw)
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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 ∈ G, is denoted by S G. We show that S G by a natural binary operation is a monoid. S G (α), the group of units in S G precisely consists of those f ∈ S G such that the map x → x f (x) is a bijection on G. Similar to the group of bisections, S G (α) 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 (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 G 2 is isomorphic to the group S G (α) and the group of transitive bisections of G, Bis T (G), is embedded in S G (α), where G 2 is the groupoid of all composable pairs.
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