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 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 \(\mathcal{SIF}^1{\times}\mathcal{SIF}^1\) where \(\mathcal{SIF}^1\) is the set of all shift-invariant filters on \(\omega\) with an attached element \(1\) endowed with the following partial order: \(\mathcal{F}\leq \mathcal{G}\) if and only if \(\mathcal{G}=1\) or \(\mathcal{F}\subset \mathcal{G}\). Also, we investigate cardinal characteristics of the lattice \(\mathcal{W}\). In particular, we prove 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 \(\mathfrak{t}\).