Lviv Topological Algebra Seminar

Scientific Seminar:

Topological Algebra

Speaker:	O. Gutik (National University of L'viv)
Talk:	On monoids of injective partial cofinite selfmaps.
When & where:	April 4, 2012 at 13 ³⁰ at Room 149^à
Abstract:	We discuss on the semigroup $\mathscr{I}^{\mathrm{cf}}_\lambda$ of injective partial cofinite selfmaps of infinite cardinal $\lambda$. We show that $\mathscr{I}^{\mathrm{cf}}_\lambda$ is a bisimple inverse semigroup and for every non-empty chain $L$ in $E(\mathscr{I}^{\mathrm{cf}}_\lambda)$ there exists an inverse subsemigroup $S$ of $\mathscr{I}^{\mathrm{cf}}_\lambda$ such that $S$ is isomorphic to the bicyclic semigroup and $L\subseteq E(S)$, we describe the Green relations on $\mathscr{I}^{\mathrm{cf}}_\lambda$ and we prove that every non-trivial congruence on $\mathscr{I}^{\mathrm{cf}}_\lambda$ is a group congruence. We also prove that every Hausdorff locally compact topology $\tau$ on $\mathscr{I}^{\mathrm{cf}}_\lambda$ such that $(\mathscr{I}^{\mathrm{cf}}_\lambda,\tau)$ is a semitopological semigroup, is discrete and we describe the closure of the discrete semigroup $\mathscr{I}^{\mathrm{cf}}_\lambda$ in a topological semigroup. Finally, we show that the (discrete) semigroup $\mathscr{I}^{\mathrm{cf}}_\lambda$ cannot embed into a compact-like topological semigroup for any infinite cardinal $\lambda$, and we construct two non-discrete Hausdorff topologies which turn $\mathscr{I}^{\mathrm{cf}}_\lambda$ into a topological inverse semigroup.