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

Abstract

A Hausdorff topology τ on the bicyclic monoid with adjoined zero C⁰ is called weak if it is contained in the coarsest inverse semigroup topology on C⁰. We show that the lattice W of all weak shift-continuous topologies on C⁰ is isomorphic to the lattice SIF¹×SIF¹ where SIF¹ is the set of all shift-invariant filters on ! with an attached element 1 endowed with the following partial order: F ≤ G if and only if G = 1 or F ⊂ G. Also, we investigate cardinal characteristics of the lattice W. In particular, we prove that W contains an antichain of cardinality 2ᶜ and a well-ordered chain of cardinality c. Moreover, there exists a well-ordered chain of first-countable weak topologies of order type t.