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.