A homological lower bound for order dimension of lattices (Q1568669)

The order dimension of a finite poset \(P\) is the smallest number \(d\) such that the order relation on \(P\) is the intersection of \(d\) linear orders. Equivalently, \(P\) can be embedded as an induced subposet in the cartesian product of \(d\) linear orders. In this note the authors prove that, for a finite lattice \(L\), the order dimension is bounded above by the homological dimension of the proper part \(L^\circ\) of \(L\), plus two. By the homological dimension of a poset \(P\) we mean the homological dimension of the order complex, the simplicial complex of linearly ordered subsets of \(P\). This bound had previously been established for face lattices of complex polytopes. A simple example shows that this upper bound can fail miserably for posets which are not lattices. The proof uses the following lemma: if the lattice \(L\) embeds as a join-sublattice of the cartesian product of \(d\) linear orders, then the order complex of \(L^\circ\) is homotopy equivalent to a complex with at most \(d\) vertices. It follows immediately that \(L^\circ\) has homological dimension at most \(d-2\). The proof of the lemma is a nice application of the Crosscut Theorem. The main theorem is proved by first reducing to the case where \(L\) is generated by atoms, embedding \(L\) in a larger lattice to which the lemma applies, and then using a Mayer-Vietoris argument and some explicit computation to show that the homological dimension of \(L^\circ\) does not increase.