Homotopy type of some lattices with top and bottom removed

Let $C(P)$ be the poset obtained by removing top and bottom from $P$ (where $P$ is a poset having a top and a bottom, not equal). Then $C(P\times Q)$ is homotopy equivalent to $\Sigma(C(P)\ast C(Q))$, the suspension of the join. Thus if one of the linear orders in your product has at least three elements then $C$ of the product will indeed be contractible, since the join of $X$ with a contractible space is always contractible. Note that the join of $X$ with the empty space is $X$.