On compact topologies on the semigroup of finite partial order isomorphisms of a bounded rank of an infinite linear ordered set

TL;DR

This paper investigates the topological properties of the semigroup of finite partial order isomorphisms on an infinite linear order, establishing uniqueness and compactness of certain topologies and their separation properties.

Contribution

It characterizes the topological structures on the semigroup of finite partial order isomorphisms, proving uniqueness of compact topologies and describing their separation and compactness properties.

Findings

Every $T_1$ left- or right-topological semigroup is completely Hausdorff and totally separated.

There exists a unique Hausdorff countably compact (pseudocompact) shift-continuous topology on the semigroup, which is compact.

The Bohr compactification of the semigroup is trivial.

Abstract

We study topologization of the semigroup $\mathscr{O\!\!I}\!_n(L)$ of finite partial order isomorphisms of a bounded rank of an infinite linear ordered set $(L,\leqslant)$. In particular we show that every $T_1$ left-topological (right-topological) semigroup $\mathscr{O\!\!I}\!_n(L)$ is a completely Hausdorff, Urysohn, totally separated, scattered space. We prove that on the semigroup $\mathscr{O\!\!I}\!_n(L)$ admits a unique Hausdorff countably compact (pseudocompact) shift-continuous topology which is compact, and the Bohr compactification of a Hausdorff topological semigroup $\mathscr{O\!\!I}\!_n(L)$ is the trivial semigroup.