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

In this paper we study the semigroup IC(I,[a]) (IO(I,[a])) of closed (open) connected partial homeomorphisms of the unit interval I with a fixed point a∈I. We describe left and right ideals of IC(I,[0]) and the Green's relations on IC(I,[0]). We show that the semigroup IC(I,[0]) is bisimple and every non-trivial congruence on IC(I,[0]) is a group congruence. Also we prove that the semigroup IC(I,[0]) is isomorphic to the semigroup IO(I,[0]) and describe the structure of a semigroup II(I,[0])=IC(I,[0])⊔IO(I,[0]). As a corollary we get structures of semigroups IC(I,[a]) and IO(I,[a]) for an interior point a∈I

On H-closed topological semigroups and semilattices

In this paper, we show that if S is an H-closed topological semigroup and e is an idempotent of S, then eSe is an H-closed topological semigroup. We give sufficient conditions on a linearly ordered topological semilattice to be H-closed. Also we prove that any H-closed locally compact topological semilattice and any H-closed topological weakly U-semilattice contain minimal idempotents. An example of a countably compact topological semilattice whose topological space is H-closed is constructed

On chains in HH-closed topological pospaces

We study chains in an $H$-closed topological partially ordered space. We give sufficient conditions for a maximal chain $L$ in an $H$-closed topological partially ordered space such that $L$ contains a maximal (minimal) element. Also we give sufficient conditions for a linearly ordered topological partially ordered space to be $H$-closed. We prove that any $H$-closed topological semilattice contains a zero. We show that a linearly ordered $H$-closed topological semilattice is an $H$-closed topological pospace and show that in the general case this is not true. We construct an example an $H$-closed topological pospace with a non-$H$-closed maximal chain and give sufficient conditions that a maximal chain of an $H$-closed topological pospace is an $H$-closed topological pospace.Comment: We have rewritten and substantially expanded the manuscrip

Topological monoids of almost monotone injective co-finite partial selfmaps of the set of positive integers

In this paper we study the semigroup $\mathcal{I}_{\,\infty}^{?\nearrow}(\mathbb{N})$ of partial co-finite almost monotone bijective transformations of the set of positive integers $\mathbb{N}$. We show that the semigroup $\mathcal{I}_{\,\infty}^{?\nearrow}(\mathbb{N})$ has algebraic properties similar to the bicyclic semigroup: it is bisimple and all of its non-trivial group homomorphisms are either isomorphisms or group homomorphisms. Also we prove that every Baire topology $\tau$ on $\mathcal{I}_{\,\infty}^{?\nearrow}(\mathbb{N})$ such that $(\mathcal{I}_{\infty}^{\,?\nearrow}(\mathbb{N}),\tau)$ is a semitopological semigroup is discrete, describe the closure of $(\mathcal{I}_{\infty}^{\,?\nearrow}(\mathbb{N}),\tau)$ in a topological semigroup and construct non-discrete Hausdorff semigroup topologies on $\mathcal{I}_{\infty}^{\,?\nearrow}(\mathbb{N})$