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\documentclass[12pt]{minimal}
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\begin{document}$$x\in G$$\end{document}, is denoted by SG\documentclass[12pt]{minimal}
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\begin{document}$$S_G$$\end{document}. We show that SG\documentclass[12pt]{minimal}
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\begin{document}$$S_G$$\end{document} by a natural binary operation is a monoid. SG(α)\documentclass[12pt]{minimal}
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\begin{document}$$S_G(\alpha )$$\end{document}, the group of units in SG\documentclass[12pt]{minimal}
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\begin{document}$$S_G$$\end{document} precisely consists of those f∈SG\documentclass[12pt]{minimal}
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\begin{document}$$f\in S_G$$\end{document} such that the map x↦xf(x)\documentclass[12pt]{minimal}
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\begin{document}$$x\mapsto xf(x)$$\end{document} is a bijection on G. Similar to the group of bisections, SG(α)\documentclass[12pt]{minimal}
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\begin{document}$$S_G(\alpha )$$\end{document} 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)\documentclass[12pt]{minimal}
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\begin{document}$$S_G(\alpha )$$\end{document} 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 G2\documentclass[12pt]{minimal}
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\begin{document}$$G^2$$\end{document} is isomorphic to the group SG(α)\documentclass[12pt]{minimal}
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\begin{document}$$S_G(\alpha )$$\end{document} and the group of transitive bisections of G, BisT(G)\documentclass[12pt]{minimal}
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\begin{document}$$Bis_T(G)$$\end{document}, is embedded in SG(α)\documentclass[12pt]{minimal}
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\begin{document}$$S_G(\alpha )$$\end{document}, where G2\documentclass[12pt]{minimal}
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\begin{document}$$G^2$$\end{document} is the groupoid of all composable pairs.
