# Positive answers to Koch's problem in special cases

## TL;DR

This paper investigates special cases of Koch's problem in topological monoids, establishing conditions under which locally compact monothetic monoids are actually compact groups, thus advancing understanding in topological algebra.

## Contribution

It provides new positive results for Koch's problem in specific classes of topological monoids, linking properties like open shifts and non-viscosity to compactness and group structure.

## Key findings

- A locally compact monothetic topological monoid is a compact topological group under certain conditions.
- Equivalence of being a submonoid of a quasitopological group, having open shifts, and being non-viscous.
- Extension of positive results in Koch's problem to special classes of topological monoids.

## 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 $a\in S$ there exists a neighborhood $V$ of $a$ such that any element $x\in S$ with $(xV\cup Vx)\cap V\ne\emptyset$ belongs to the neighborhood $U$ of 1.