Infinite dual symmetric inverse monoids (Q1705830)

Here, the dual symmetric inverse monoid \(\mathcal{I}_X^*\) on an infinite set \(X\) is studied, i.e. the semigroup of all bijections between quotients of \(X\). If \({\mathcal{S}_X}\) is the symmetric group, then the relative rank (Sierpinski rank) \({\text{rank}}(\mathcal{I}_X^*:{\mathcal{S}_X}) = 2\). From this, it follows that the semigroup \(\mathcal{I}_X^*\) can be generated by the symmetric group \({\mathcal{S}_X}\) together with two elements \(\alpha ,\beta \in \mathcal{I}_X^*\). The characterization of all such pairs \(\alpha ,\beta \) depends on whether the cardinal \(| X| \) is regular or singular. If \({\mathcal{E}_X}\) is the set of idempotents of \(\mathcal{I}_X^*\) and \(\mathcal{I}_X^*\) is generated by a set \({\mathcal{E}_X} \cup \Sigma \), \(\Sigma \subseteq \mathcal{I}_X^*\), then \(| \Sigma | \geqslant {2^{| X|}}\) and \(\mathcal{I}_X^*\) is generated by the set \(\Sigma \) alone. The semigroup \(\mathcal{I}_X^*\) also has the semigroup Bergman property, i.e. any countable subset of \(\mathcal{I}_X^*\) is contained in a 4-generator subsemigroup.