Semigroup Closures of Finite Rank Symmetric Inverse Semigroups

TL;DR

This paper studies the algebraic closure properties of finite rank symmetric inverse semigroups within semitopological inverse semigroups, revealing their algebraic completeness and limitations on compactifications.

Contribution

It introduces the concept of semigroup with a tight ideal series and proves finite rank symmetric inverse semigroups are algebraically closed in certain topological classes.

Findings

Finite rank symmetric inverse semigroups are algebraically closed in semitopological inverse semigroups.

No partial compactifications exist for the classes of semigroups considered.

The notion of semigroup with a tight ideal series is introduced and studied.

Abstract

We introduce the notion of semigroup with a tight ideal series and investigate their closures in semitopological semigroups, particularly inverse semigroups with continuous inversion. As a corollary we show that the symmetric inverse semigroup of finite transformations $\mathscr{I}_\lambda^n$ of the rank $\leqslant n$ is algebraically closed in the class of (semi)topological inverse semigroups with continuous inversion. We also derive related results about the nonexistence of (partial) compactifications of classes of semigroups that we consider.