Discrete Morse theory for the collapsibility of supremum sections
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The Dushnik-Miller dimension of a poset < is the minimal number d of linear extensions <_1, ... , <_d of < such that < is the intersection of <_1, ... , <_d. Supremum sections are simplicial complexes introduced by Scarf and are linked to the Dushnik-Miller as follows: the inclusion poset of a simplicial complex is of Dushnik-Miller dimension at most d if and only if it is included in a supremum section coming from a representation of dimension d. Collapsibility is a topoligical property of simplicial complexes which has been introduced by Whitehead and which resembles to shellability. While Ossona de Mendez proved in that a particular type of supremum sections are shellable, we show in this article that supremum sections are in general collapsible thanks to the discrete Morse theory developped by Forman.
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