Infinite dual symmetric inverse monoids

Let \({\mathcal {I}}^*_X\) be the dual symmetric inverse monoid on an infinite set X. We show that \({\mathcal {I}}^*_X\) may be generated by the symmetric group \({\mathcal {S}}_X\) together with two (but no fewer) additional block bijections, and we classify the pairs \(\alpha ,\beta \in {\mathcal {I}}^*_X\) for which \({\mathcal {I}}^*_X\) is generated by \({\mathcal {S}}_X\,\cup \,\{\alpha ,\beta \}\). Among other results, we show that any countable subset of \({\mathcal {I}}^*_X\) is contained in a 4-generated subsemigroup of \({\mathcal {I}}^*_X\), and that the length function on \({\mathcal {I}}^*_X\) (and its finitary power semigroup) is bounded with respect to any generating set.