The MacNeille completion of the poset of partial injective functions (Q1010782)

Summary: Renner has defined an order on the set of partial injective functions from \([n]=\{1,\dots,n\}\) to \([n]\). This order extends the Bruhat order on the symmetric group. The poset \(P_{n}\) obtained is isomorphic to a set of square matrices of size \(n\) with its natural order. We give the smallest lattice that contains \(P_{n}\). This lattice is in bijection with the set of alternating matrices. These matrices generalize the classical alternating sign matrices. The set of join-irreducible elements of \(P_{n}\) are increasing functions for which the domain and the image are intervals.