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

Abstract

A Hausdorff topology $\tau$ on the bicyclic monoid with adjoined zero ${\mathcal{C}}^0$ is called {\em weak} if it is contained in the coarsest inverse semigroup topology on ${\mathcal{C}}^0$. We show that the lattice $\mathcal{W}$ of all weak shift-continuous topologies on ${\mathcal{C}}^0$ is isomorphic to the lattice of all shift-invariant filters on $\omega$ with an attached element $1$ endowed with the following partial order: $\mathcal{F}\le \mathcal{G}$ iff $\mathcal{G}=1$ or $\mathcal{F}\subset \mathcal{G}$. Also, we investigate cardinal characteristics of the lattice $\mathcal{W}$. In particular, we proved that $\mathcal{W}$ contains an antichain of cardinality $2^{\mathfrak{c}}$ and a well-ordered chain of cardinality $\mathfrak{c}$. Moreover, there exists a well-ordered chain of first-countable weak topologies of order type $\mathfrak{t}$.

References in corpus (4)

• On a complete topological inverse polycyclic monoid
• Classifying locally compact semitopological polycyclic monoids
• On the dichotomy of a locally compact semitopological bicyclic monoid with adjoined zero
• On the dichotomy of a locally compact semitopological monoid of order isomorphisms between principal filters of ${\mathbb{N}}^n$ with adjoined zero