On a New Approach to the Dual Symmetric Inverse Monoid IX*

We construct the inverse partition semigroup IPX , isomorphic to the dual symmetric inverse monoid I∗ X , introduced in [6]. We give a convenient geometric illustration for elements of IPX . We describe all maximal subsemigroups of IPX and find a generating set for IPX when X is finite. We prove that all the automorphisms of IPX are inner. We show how to embed the symmetric inverse semigroup into the inverse partition one. For finite sets X, we establish that, up to equivalence, there is a unique faithful effective transitive representation of IPn, namely to IS2n−2. Finally, we construct an interesting H-cross-section of IPn, which is reminiscent of IOn, the H-cross-section of ISn, constructed in [4].