Order complex

Remarks

$\Delta(P)$ is a simplicial complex because every subset of a chain is a chain. A chain with $k + 1$ elements is a face of dimension $k$, so the dimension of $\Delta(P)$ is the length of the longest chain of $P$, when it is finite. Topological properties of $P$, such as its homotopy type or homology, mean those of the geometric realization of $\Delta(P)$. Example: let $P = \set{a, b, c, d}$ with $a < c$, $a < d$, $b < c$, $b < d$ and no other strict relations. Its nonempty chains are the four points and the four pairs $\set{a, c}$, $\set{a, d}$, $\set{b, c}$, $\set{b, d}$. So $\Delta(P)$ is a cycle of four edges, and its realization is a circle. Non-example: for this $P$, the family of all subsets of $P$ is a simplicial complex on the vertex set $P$. It is not $\Delta(P)$, since it contains $\set{a, b}$, which is not a chain. The order complex of a chain with $n + 1$ elements is a full $n$-simplex.