Topological Algebra and its Applications (Apr 2018)
On locally compact shift-continuous topologies on the α-bicyclic monoid
Abstract
A topology τ on a monoid S is called shift-continuous if for every a, b ∈ S the two-sided shift S → S, x ↦ axb, is continuous. For every ordinal α ≤ ω, we describe all shift-continuous locally compact Hausdorff topologies on the α-bicyclic monoid Bα. More precisely, we prove that the lattice of shift-continuous locally compact Hausdorff topologies on Bα 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 Bα+1 is isomorphic to the Bruck extension of the α-bicyclic monoid Bα.
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