On semitopological bicyclic extensions of linearly ordered groups

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.