The MacNeille Completion of the Poset of Partial Injective Functions

  • Marc Fortin


Renner has defined an order on the set of partial injective functions from $[n]=\{1,\ldots,n\}$ 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.

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