On the lattice of weak topologies on the bicyclic monoid with adjoined zero

On the dichotomy of a locally compact semitopological bicyclic monoid with adjoined zero

We prove that a Hausdorff locally compact semitopological bicyclic semigroup with adjoined zero $\mathscr{C}^0$ is either compact or discrete. Also we show that the similar statement holds for a locally compact semitopological bicyclic semigroup with an adjoined compact ideal and construct an example which witnesses that a counterpart of the statements does not hold when $\mathscr{C}^0$ is a Čech-complete metrizable topological inverse semigroup.

On a semitopological extended bicyclic semigroup with adjoined zero

In the paper it is shown that every Hausdorff locally compact semigroup topology on the extended bicyclic semigroup with adjoined zero $\mathscr{C}_{\mathbb{Z}}^{\mathbf{0}}$ is discrete, but on $\mathscr{C}_{\mathbb{Z}}^{\mathbf{0}}$ there exist $\mathfrak{c}$ many different Hausdorff locally compact shift-continuous topologies. Also, it is constructed on $\mathscr{C}_{\mathbb{Z}}^{\mathbf{0}}$ the unique minimal shift continuous topology and the unique minimal inverse semigroup topology.

On the dichotomy of a locally compact semitopological monoid of order isomorphisms between principal filters of $\mathbb{N}^n$ with adjoined zero

Let $n$ be any positive integer and $\mathscr{I\!P\!F}(\mathbb{N}^n)$ be the semigroup of all order isomorphisms between principal filters of the $n$-th power of the set of positive integers $\mathbb{N}$ with the product order. We prove that a Hausdorff locally compact semitopological semigroup ${\mathscr{I\!P\!F}(\mathbb{N}^n)}$ with an adjoined zero is either compact or discrete.

On locally compact shift-continuous topologies on the $α$-bicyclic monoid

A topology $τ$ on a monoid $S$ is called {\em shift-continuous} if for every $a,b\in S$ the two-sided shift $S\to S$, $x\mapsto axb$, is continuous. For every ordinal $α\le ω$, we describe all shift-continuous locally compact Hausdorff topologies on the $α$-bicyclic monoid $\mathcal{B}_α$. More precisely, we prove that the lattice of shift-continuous locally compact Hausdorff topologies on $\mathcal{B}_α$ is anti-isomorphic to the segment of $[1,α]$ of ordinals, endowed with the natural well-order. Also we prove that for each ordinal $α$ the $α+1$-bicyclic monoid $\mathcal{B}_{α+1}$ is isomorp…

On semitopological $α$-bicyclic monoid

In this paper we consider a semitopological $α$-bicyclic monoid $\mathcal{B}_α$ and prove that it is algebraically isomorphic to a semigroup of all order isomorphisms between the principal upper sets of the ordinal $ω^α$. We prove that for every ordinal $α$ for every $(a,b)\in \mathcal{B_α}$ if either $a$ or $b$ is a non-limit ordinal then $(a,b)$ is an isolated point in $\mathcal{B}_α$. We show that for every ordinal $α<ω+1$ every locally compact semigroup topology on $\mathcal{B}_α$ is discrete. However, we construct an example of a non-discrete locally compact topology $τ_{lc}$ on $\mathcal…

关于双环单形子半群的拓扑化

On topologization of subsemigroups of the bicyclic monoid

研究证明双环单形子半群在包含无限多个幂等元时只能具有离散的Hausdorff移位连续拓扑，并且所有右连续或左连续的Hausdorff Baire拓扑均为离散。