On a semigroup of closed connected partial homeomorphisms of the unit interval with a fixed point

Abstract

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 \(a\in I\).