When a locally compact monothetic semigroup is compact

A semigroup endowed with a topology is monothetic if it contains a dense monogenic subsemigroup. A semigroup (group) endowed with a topology is semitopological (quasitopological) if the translations (the translations and the inversion) are continuous. If S is a nondiscrete monothetic semitopological semigroup, then the set {S^{\prime}} of all limit points of S is a closed ideal of S. Let S be a locally compact nondiscrete monothetic semitopological semigroup. We show that (1) if the translations of {S^{\prime}} are open, then {S^{\prime}} is compact, and (2) if {S^{\prime}} can be topologically and algebraically embedded in a quasitopological group, then {S^{\prime}} is a compact topological group.

Positive answers to Koch’s problem in special cases

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 S 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 ∪ Vx) ∩ V ≠ ∅ belongs to the neighborhood U of 1.