Embedding the bicyclic semigroup into countably compact topological semigroups

Abstract

We study algebraic and topological properties of topological semigroups containing a copy of the bicyclic semigroup C(p,q). We prove that each topological semigroup S with pseudocompact square contains no dense copy of C(p,q). On the other hand, we construct a (consistent) example of a pseudocompact (countably compact) Tychonov semigroup containing a copy of C(p,q).

14 pages

References in corpus (3)

• On countably compact 0-simple topological inverse semigroups
• The continuity of the inversion and the structure of maximal subgroups in countably compact topological semigroups
• The Rees-Suschkewitsch Theorem for simple topological semigroups

Cited by in corpus (16)

• On monoids of monotone injective partial self-maps of integers with cofinite domains and images
• On inverse submonoids of the monoid of almost monotone injective co-finite partial selfmaps of positive integers
• The continuity of the inversion and the structure of maximal subgroups in countably compact topological semigroups
• On locally compact semitopological $0$-bisimple inverse $\omega$-semigroups
• Openly factorizable spaces and compact extensions of topological semigroups
• Suitable sets for paratopological groups
• Topological monoids of monotone injective partial selfmaps of $\mathbb{N}$ with cofinite domain and image
• Topological monoids of almost monotone injective co-finite partial selfmaps of positive integers
• On the dichotomy of a locally compact semitopological monoid of order isomorphisms between principal filters of ${\mathbb{N}}^n$ with adjoined zero
• On the monoid of cofinite partial isometries of $\mathbb{N}$ with a bounded finite noise
• On the closure of the extended bicyclic semigroup
• On the lattice of weak topologies on the bicyclic monoid with adjoined zero
• Toplogical semigroups embedded into topological groups
• On the semigroup ${\text{ID}}_{\infty }$
• On some generalization of the bicyclic semigroup: the topological version
• The monoid of monotone injective partial selfmaps of the poset $({\mathbb{N}}^3,⩽)$ with cofinite domains and images