Order complexes of noncomplemented lattices are nonevasive

Abstract

We prove the following theorem: Let L \mathcal {L} be a finite lattice, x ∈ L ¯ x\in \bar {\mathcal {L}} . Assume B B is a set of elements of L \mathcal {L} which includes all complements of x x and is included in the set of all upper (lower) semicomplements of x x . Then δ ( L ∖ B ¯ ) \delta (\overline {\mathcal {L}\setminus B}) is nonevasive, in particular collapsible. This generalizes results of several previous papers, where, in different generalities, it has been proved that the mentioned complex is contractible.