Geodesic
On a semigroup of closed connected partial homeomorphisms of the unit interval with a~fixed point
Algebra and discrete mathematics, Tome 12 (2011) no. 2, pp. 38-52.

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In this paper we study the semigroup $\mathfrak{IC}(I,[a])$ ($\mathfrak{IO}(I,[a])$) of closed (open) connected partial homeomorphisms of the unit interval $I$ with a fixed point $a\in I$. We describe left and right ideals of $\mathfrak{IC}(I,[0])$ and the Green's relations on $\mathfrak{IC}(I,[0])$. We show that the semigroup $\mathfrak{IC}(I,[0])$ is bisimple and every non-trivial congruence on $\mathfrak{IC}(I,[0])$ is a group congruence. Also we prove that the semigroup $\mathfrak{IC}(I,[0])$ is isomorphic to the semigroup $\mathfrak{IO}(I,[0])$ and describe the structure of a semigroup $\mathfrak{II}(I,[0])=\mathfrak{IC}(I,[0])\sqcup\mathfrak{IO}(I,[0])$. As a corollary we get structures of semigroups $\mathfrak{IC}(I,[a])$ and $\mathfrak{IO}(I,[a])$ for an interior point $a\in I$.
Keywords: Semigroup of bijective partial transformations, symmetric inverse semigroup, semigroup of homeomorphisms, bisimple semigroup.
Mots-clés : group congruence 
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Ivan Chuchman. On a semigroup of closed connected partial homeomorphisms of the unit interval with a~fixed point. Algebra and discrete mathematics, Tome 12 (2011) no. 2, pp. 38-52. http://geodesic.mathdoc.fr/item/ADM_2011_12_2_a3/

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