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On semigroups which admit only discrete left-continuous Hausdorff topology

Oleg Gutik

TL;DR

The paper investigates when every Hausdorff left-continuous (or right-continuous) topology on a semigroup must be discrete, focusing on the bicyclic monoid ${\mathscr{C}}(a,b)$ and its extensions. It provides a general sufficient condition ensuring discreteness of such topologies and constructs continuum-many subsemigroups ${S_\alpha}$ inside ${\mathscr{C}}_{+}(a,b)$ and ${\mathscr{C}}_{-}(a,b)$ that simultaneously admit non-discrete topologies on the opposite side and embed into compact Hausdorff semigroups. It also extends these phenomena to subsemigroups of ${\mathscr{C}}^{+}_{\mathbb{Z}}$ and ${\mathscr{C}}^{-}_{\mathbb{Z}}$, with anti-isomorphic duals, and analyzes non-discrete shift-continuous topologies on the extended semigroup ${\mathscr{C}}_{\mathbb{Z}}$. Finally, the paper describes compactification constructions that place ${\mathscr{C}}_{\mathbb{Z}}^{+}$ (and analogues) densely inside compact topological semigroups, ensuring that every subsemigroup has a dense compactification, and discusses the broader landscape of shift-continuous locally compact topologies for these algebras.

Abstract

We give the sufficient condition when every left-continuous (right-continuous) Hausdorff topology on a semigroup $S$ is discrete. We construct a submonoid $\mathscr{C}_{+}(a,b)$ (resp., $\mathscr{C}_{-}(a,b)$) of the bicyclic monoid which contains a family $\{S_α\colon α\in\mathfrak{c}\}$ of continuum many subsemigroups with the following properties: $(i)$ every left-continuous (resp., right-continuous) Hausdorff topology on $S_α$ is discrete; $(ii)$ every semigroup $S_α$ admits a non-discrete right-continuous (resp., left-continuous) Hausdorff topology which is not left-continuous (resp., right-continuous); $(iii)$ every semigroup $S_α$ isomorphically embeds into a Hausdorff compact topological semigroup. Also we construct a submonoid $\mathscr{C}_{\mathbb{Z}}^+$ (resp., $\mathscr{C}_{\mathbb{Z}}^-$) of the extended bicyclic semigroup which contains a family $\{S_α\colon α\in\mathfrak{c}\}$ of continuum many subsemigroups with the above described properties.

On semigroups which admit only discrete left-continuous Hausdorff topology

TL;DR

The paper investigates when every Hausdorff left-continuous (or right-continuous) topology on a semigroup must be discrete, focusing on the bicyclic monoid and its extensions. It provides a general sufficient condition ensuring discreteness of such topologies and constructs continuum-many subsemigroups inside and that simultaneously admit non-discrete topologies on the opposite side and embed into compact Hausdorff semigroups. It also extends these phenomena to subsemigroups of and , with anti-isomorphic duals, and analyzes non-discrete shift-continuous topologies on the extended semigroup . Finally, the paper describes compactification constructions that place (and analogues) densely inside compact topological semigroups, ensuring that every subsemigroup has a dense compactification, and discusses the broader landscape of shift-continuous locally compact topologies for these algebras.

Abstract

We give the sufficient condition when every left-continuous (right-continuous) Hausdorff topology on a semigroup is discrete. We construct a submonoid (resp., ) of the bicyclic monoid which contains a family of continuum many subsemigroups with the following properties: every left-continuous (resp., right-continuous) Hausdorff topology on is discrete; every semigroup admits a non-discrete right-continuous (resp., left-continuous) Hausdorff topology which is not left-continuous (resp., right-continuous); every semigroup isomorphically embeds into a Hausdorff compact topological semigroup. Also we construct a submonoid (resp., ) of the extended bicyclic semigroup which contains a family of continuum many subsemigroups with the above described properties.
Paper Structure (4 sections, 18 theorems, 29 equations)

This paper contains 4 sections, 18 theorems, 29 equations.

Table of Contents

  1. Introduction, motivation and main definitions
  2. On topologizations of subsubsemigroups of the semigroups $\mathscr{C}_{+}(a,b)$ and $\mathscr{C}_{-}(a,b)$
  3. On topologizations of subsubsemigroups of the semigroups $\mathscr{C}^{+}_{\mathbb{Z}}$ and $\mathscr{C}^{-}_{\mathbb{Z}}$
  4. Embedding of the semigroup ${\mathscr{C}}_\mathbb{Z}^+$ into a compact topological semigroup

Key Result

Theorem 2.2

Let $S$ be an infinite semigroup. If for any $s\in S$ there exists an idempotent $e_s\in S$ such that $s\in S\setminus Se_s$$(s\in S\setminus e_sS)$ and the set $S\setminus Se_s$$(S\setminus e_sS)$ is finite, then every Hausdorff left-contionuous (right-continuous) topology on $S$ is discrete.

Theorems & Definitions (41)

  • Definition 1.1
  • Definition 1.2: Carruth-Hildebrant-Koch=1983Ruppert=1984
  • Example 2.1
  • Theorem 2.2
  • proof
  • Corollary 2.3
  • proof
  • Example 2.4
  • Lemma 2.5
  • proof
  • ...and 31 more