On a semitopological polycyclic monoid

Abstract

We study algebraic structure of the \(\lambda\)-polycyclic monoid \(P_{\lambda}\) and its topologizations. We show that the \(\lambda\)-polycyclic monoid for an infinite cardinal \(\lambda\geqslant 2\) 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 \(\lambda\) the polycyclic monoid \(P_{\lambda}\) 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 \((P_{\lambda},\tau)\) for every Hausdorff topology \(\tau\) on \(P_{\lambda}\), such that \((P_{\lambda},\tau)\) is a semitopological semigroup, and every locally compact Hausdorff semigroup topology on \(P_\lambda\) is discrete. The last statement extends results of the paper [33] obtaining for topological inverse graph semigroups. We describe all feebly compact topologies \(\tau\) on \(P_{\lambda}\) such that \(\left(P_{\lambda},\tau\right)\) is a semitopological semigroup and its Bohr compactification as a topological semigroup. We prove that for every cardinal \(\lambda\geqslant 2\) any continuous homomorphism from a topological semigroup \(P_\lambda\) into an arbitrary countably compact topological semigroup is annihilating and there exists no a Hausdorff feebly compact topological semigroup which contains \(P_{\lambda}\) as a dense subsemigroup.