A note on the homotopy type of posets

AbstractFor 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 Jl(f)(resp.Jr(f)). Our first result asserts that if f:P→Q,g:Q→P are maps of a Galois connection, then (a) Jr(f):J(Q∗)→J(P)∗,Jl(g):J(P∗)→J(Q∗) and (b) Jl(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 (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)