Semigroup closures of finite rank symmetric inverse semigroups (Q1014265)

The authors call a subset \(D\) of a semigroup \(S\) \(\omega\)-unstable if \(D\) is infinite and, for any \(a\in D\) and an infinite subset \(B\) of \(D\), one has \(aB\cup Ba\nsubseteq D\). A chain \(I_0\subseteq I_1\subseteq \dots\subseteq I_m\) of ideals of \(S\) is called a tight series if \(I_0\) is finite, \(I_m=S\) and \(I_k\setminus I_{k-1}\) is \(\omega\)-unstable for each \(k\in\{1,\dots,m\}\). If a semitopological regular semigroup \(S\) has a tight series of ideals, then each \(I_k\) is closed and each point from \(S\setminus I_{m-1}\) is isolated. Using this and similar observations, the authors show that, for every infinite cardinal \(\lambda\) and every natural \(n\), the semigroup \(\mathcal{P}_{\lambda}^n\) of all partial bijections of \(\lambda\) of \(\text{rank}\leqslant n\) is closed in every topological semigroup containing \(\mathcal{P}_{\lambda}^n\).