A note on the homotopy type of posets

Abstract

For any poset P let J(P) denote the complete lattice of order ideals in P. J(P) is a contravariant functor in P. Any order-reserving map f:P→Q can be regarded as an isotone (=order-preserving) map of either P^∗ into Q or P into Q^∗. The induced map of J(Q) to J(P^∗)(resp. J(Q^∗) into J(P)) will be denoted by J_l(f)(resp.J_r(f)). Our first result asserts that if f:P→Q,g:Q→P are maps of a Galois connection, then (a) J_r(f):J(Q^∗)→J(P)^∗,Jl(g):J(P^∗)→J(Q^∗) and (b) J_l(f):J(Q)^∗→J(P^∗),Jr(g):J(P^∗)→J(Q)^∗ are Galois connections. For any lattice L, we denote the poset L - {0,1} by L̄. We analyse conditions which will imply that Jr(f)(J (Q^∗)) ∋J (P)^∗ and Jl(g)(J (P)^∗) ∋ J(Q)∗. Under these conditions, from Walker's results [3] it will follow that Jr(f)/(J (Q^∗))→(J (P)^∗ is a homotopy equivalence with Jl(g)(J (P)∗(J (P)^∗:→(J (Q^∗))as its homotopy inverse. Given an isotone map f:P→Q it is easy to find the necessary and sufficient conditions for J(f) to satisfy J(f)(J(Q))⊂J(P). When these conditions are fulfilled, we also find a sufficient condition that ensures that J(f)/J(Q):J(Q)→J(P)is a homotopy equivalence. We give examples to show that the homotopy type of P neither determines nor is determined by the homotopy type of J (P).