Q:

Poset complex of reverse ordering

This might be too easy but I cannot proof it easily. Any reference or hint will be great. 
Q: Suppose P is a poset in which every chain is finite and $\Delta P$ is the poset complex associated to it. (i.e. faces of $\Delta P$ are the chains of the poset.). Suppose I reverse the order in the poset and suppose it again gives me a poset say $\bar{P}$. Let $\bar{P}$ also has the property that every chain is finite. Does $\Delta P$ and $\Delta \bar{P}$ are homeomorphic. 
Simple examples shows me that the second one gives me a subdivision of the first one, but is it true in general. Thank's in advance.

A:

Let $f:P\to \bar{P}$ is the antiisomorphism. It maps a chain $a_1<\ldots< a_n$ into $a_n<\ldots a_1$ so it maps every simplex into itself.

