ON A NEW APPROACH TO THE DUAL SYMMETRIC INVERSE MONOID $\mathcal{I}_{X}^{\ast}$

Abstract

We construct the inverse partition semigroup[Formula: see text], isomorphic to the dual symmetric inverse monoid[Formula: see text], introduced in [6]. We give a convenient geometric illustration for elements of [Formula: see text]. We describe all maximal subsemigroups of [Formula: see text] and find a generating set for [Formula: see text] when X is finite. We prove that all the automorphisms of [Formula: see text] 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 [Formula: see text], namely to [Formula: see text]. Finally, we construct an interesting [Formula: see text]-cross-section of [Formula: see text], which is reminiscent of [Formula: see text], the [Formula: see text]-cross-section of [Formula: see text], constructed in [4].