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A Group Of Continuous Self-maps On A Topological Groupoid
Published 2018 · Mathematics
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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 $$x\in G$$x∈G, is denoted by $$S_G$$SG. We show that $$S_G$$SG by a natural binary operation is a monoid. $$S_G(\alpha )$$SG(α), the group of units in $$S_G$$SG precisely consists of those $$f\in S_G$$f∈SG such that the map $$x\mapsto xf(x)$$x↦xf(x) is a bijection on G. Similar to the group of bisections, $$S_G(\alpha )$$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 $$S_G(\alpha )$$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 $$G^2$$G2 is isomorphic to the group $$S_G(\alpha )$$SG(α) and the group of transitive bisections of G, $$Bis_T(G)$$BisT(G), is embedded in $$S_G(\alpha )$$SG(α), where $$G^2$$G2 is the groupoid of all composable pairs.