On locally compact shift-continuous topologies on the α-bicyclic monoid

2017/07/22 by Serhii Bardyla, Bardyla, Serhii
Computer Science · Mathematics · #Advanced Algebra and Logic #Advanced Topology and Set Theory #FOS: Mathematics #General Topology (math.GN) #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.1707.07130

openalex publication_date 2017/07/22 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28

Abstract

A topology τ on a monoid S is called \em 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 B_α. More precisely, we prove that the lattice of shift-continuous locally compact Hausdorff topologies on B_α 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 B_α+1 is isomorphic to the Bruck extension of the α-bicyclic monoid B_α.

Related

• On the lattice of weak topologies on the bicyclic monoid with adjoined\n zero
• Classifying locally compact semitopological polycyclic monoids
• On non-topologizable semigroups
• On locally compact shift continuous topologies on the semigroup \boldsymbolB_[0,∞) with an adjoined compact ideal
• Topological Transformation Monoids