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

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 ω 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.