On semitopological bicyclic extensions of linearly ordered groups

TL;DR

This paper investigates the algebraic and topological properties of semitopological bicyclic extensions of linearly ordered groups, showing that certain topologies on these structures are necessarily discrete under specific conditions.

Contribution

It characterizes the natural partial order, solutions of equations, and topologizations of semigroups formed from shifts of positive cones in linearly ordered groups, revealing conditions for discreteness.

Findings

Baire shift-continuous T1-topologies are discrete on countable groups.

Shift-continuous Hausdorff topologies are discrete for non-densely ordered groups.

Semigroup structures are inherently discrete under these topological conditions.

Abstract

For a linearly ordered group $G$ let us define a subset $A\subseteq G$ to be a \emph{shift-set} if for any $x,y,z\in A$ with $y < x$ we get $x\cdot y^{-1}\cdot z\in A$. We describe the natural partial order and solutions of equations on the semigroup $\mathscr{B}(A)$ of shifts of positive cones of $A$. We study topologizations of the semigroup $\mathscr{B}(A)$. In particular, we show that for an arbitrary countable linearly ordered group $G$ and a non-empty shift-set $A$ of $G$ every Baire shift-continuous $T_1$-topology $\tau$ on $\mathscr{B}(A)$ is discrete. Also we prove that for an arbitrary linearly non-densely ordered group $G$ and a non-empty shift-set $A$ of $G$, every shift-continuous Hausdorff topology $\tau$ on the semigroup $\mathscr{B}(A)$ is discrete, and hence $\left(\mathscr{B}(A),\tau\right)$ is a discrete subspace of any Hausdorff semitopological semigroup which contains $\mathscr{B}(A)$ as a subsemigroup.