Geodesic
The MacNeille completion of the poset of partial injective functions
The electronic journal of combinatorics, Tome 15 (2008)
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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.
DOI : 10.37236/786
Classification : 06A11, 05C50, 06D05
Mots-clés : alternating matrix, Bruhat order, dissective, distributive lattice, join-irreducible elements 
@article{10_37236_786,
     author = {Marc Fortin},
     title = {The {MacNeille} completion of the poset of partial injective functions},
     journal = {The electronic journal of combinatorics},
     year = {2008},
     volume = {15},
     doi = {10.37236/786},
     zbl = {1192.06004},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/786/}
}
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Marc Fortin. The MacNeille completion of the poset of partial injective functions. The electronic journal of combinatorics, Tome 15 (2008). doi: 10.37236/786

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