On a semitopological polycyclic monoid

Abstract

We study algebraic structure of the $λ$-polycyclic monoid $P_λ$ and its topologizations. We show that the $λ$-polycyclic monoid for an infinite cardinal $λ\geqslant 2$ has similar algebraic properties so has the polycyclic monoid $P_n$ with finitely many $n\geqslant 2$ generators. In particular we prove that for every infinite cardinal $λ$ the polycyclic monoid $P_λ$ is a congruence-free combinatorial $0$-bisimple $0$-$E$-unitary inverse semigroup. Also we show that every non-zero element $x$ is an isolated point in $(P_λ,τ)$ for every Hausdorff topology $τ$ on $P_λ$, such that $(P_λ,τ)$ is a semitopological semigroup, and every locally compact Hausdorff semigroup topology on $P_λ$ is discrete. The last statement extends results of the paper [33] obtaining for topological inverse graph semigroups. We describe all feebly compact topologies $τ$ on $P_λ$ such that $\left(P_λ,τ\right)$ is a semitopological semigroup and its Bohr compactification as a topological semigroup. We prove that for every cardinal $λ\geqslant 2$ any continuous homomorphism from a topological semigroup $P_λ$ into an arbitrary countably compact topological semigroup is annihilating and there exists no a Hausdorff feebly compact topological semigroup which contains $P_λ$ as a dense subsemigroup.