Positive answers to Koch’s problem in special cases

Abstract

A topological semigroup is monothetic provided it contains a dense cyclic subsemigroup. The Koch problem asks whether every locally compact monothetic monoid is compact. This problem was opened for more than sixty years, till in 2018 Zelenyuk obtained a negative answer. In this paper we obtain a positive answer for Koch's problem for some special classes of topological monoids. Namely, we show that a locally compact monothetic topological monoid is a compact topological group if and only if S is a submonoid of a quasitopological group if and only if S has open shifts if and only if S is non-viscous in the sense of Averbukh. The last condition means that any neighborhood U of the identity 1 of S and for any element a ∈ S there exists a neighborhood V of a such that any element x ∈ S with (xV ∪ V x) ∩ V = ∅ belongs to the neighborhood U of 1.

On a locally compact monoid of cofinite partial isometries of $\mathbb{N}$ with adjoined zero

“…From the other side, Zelenyuk in [53] constructed a countable monothetic locally compact topological semigroup without unit which is neither compact nor discrete and in [54] he constructed a monothetic locally compact topological monoid with the same property. The topological properties of monothetic locally compact (semi)topological semigroups studied in [3,20,55,56].…”

On a semitopological semigroup $\boldsymbol{B}_{\omega}^{\mathscr{F}}$ when a family $\mathscr{F}$ consists of inductive non-empty subsets of $\omega$

“…From the other hand, Zelenyuk in [33] constructed a countable monothetic locally compact topological semigroup without an identity which is neither compact nor discrete and in [34] he constructed a monothetic locally compact topological monoid with the same property. The topological properties of monothetic locally compact (semi)topological semigroups are studied in [2,14,35,36]. In the paper [15] it is proved that every Hausdorff locally compact shift-continuous topology on the bicyclic monoid with an adjoined zero is either compact or discrete.…”

When a locally compact monothetic semigroup is compact

“…A locally compact monothetic topological semigroup without identity which is neither compact nor discrete was constructed in [6], and a full negative answer to Koch's question was provided in [7]. On the other hand, if S is a locally compact monothetic topological semigroup with identity and the translations of S are open or S can be topologically and algebraically embedded in a quasitopological group, then S is compact [1].…”