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

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 $a\in I$.