# Posets, simplicial complexes, and polytopes

### Posets

Definition partially ordered set or poset is a set $P$ equipped with a relation $\leq$ that is reflexive, antisymmetric, and transitive. That is, for all $x,y,z\in P$:

1. $x\leq x$.
2. If $x\leq y$ and $y\leq x$, then $x=y$ (antisymmetry)
3. If $x\leq y$ and $y\leq z$, then $x\leq z$ (transitivity).

We say that $x$ is covered by $y$, written $x\lessdot y$, if $x and there is no $z$ such that $x. Two posets $P,Q$ are isomorphic  if  there is a bijection $\phi:P\to Q$ that is order-persverin; that is, $x\leq y$ in $P$ iff $\phi(x)\leq \phi(y)$ in $Q$.

We’ll usually assume that $P$ is finite.

### Ranked posets

Definition A chain $x_0 is saturated if it is not properly contained in any other chain from $x_0$ to $x_n$; equivalently, if $x_{i-1}\lessdot x_i$ for every $i\in [n]$. In this case, the number $n$ is the length of the chain. A poset $P$ is ranked if for every $x\in P$, all saturated chains with top element $x$ have the same length; this number is called the rank  of $x$ and denoted $r(x)$.  A poset is graded if it is ranked and bounded.

Definition Let $P$ be a ranked set with rank function $r$. The rank-generating function  of $P$ is the formal power series

$\displaystyle F_p(q):=\sum_{x\in P}q^{r(x)}.$

### Simplicial complexes

Definition Let $V$ be a finite set of vertices. An simplicial complex $\Delta$ on $V$ is a nonempty family of subsets of $V$ with the property that if $\sigma\in\Delta$ and $\tau\subset\sigma$, then $\tau\in\Delta$. Equivalently, $\Delta$ is an order ideal in the Boolean algebra $2^V$. The element of $\Delta$ are called its faces or simplices.

### The incidence algebra of a poset

Let $P$ be a poset and let $\mathrm{Int}(P)$ denote the set of (nonempty) intervals of $P$, i.e., the sets

$\displaystyle [x,y]:=\{z\in P:x\leq z\leq y\}$

for all $x\leq y$.

We will always assume that $p$ is locally finite, i.e., every interval is finite.

Definition (Incidence algebra) The incidence algebra is the set of funciton $f:\mathrm{Int}(P)\to\mathbf{C}$ (“incidence funcitons”), made into a $\mathbf{C}$-vector space with pointwise addition, subtraction and scalar multiplication, and equipped with the convolution product

$\displaystyle (f*g)(x,y)=\sum_{z\in[x,y]}f([x,z])g([z,y]).$

It is often convenient to set $f(x,y)=0$ if $x\nleqslant y$. Note that the assumption of local finiteness is both necessary and sufficient for convolution to be well-defined.

Proposition Convolution is associative (although it is not in general commutative).

Proposition An incidence function $f\in I(P)$ has a left/right two side side-convolution inverse if and only if $f([x,x])\neq 0$ for all $x$. In that case, the inverse is given by the formula

$\displaystyle f^{-1}([x,y])=\begin{cases} f([x,x])^{-1} & \text{if}\ x=y,\\ -f([y,y])^{-1}\sum_{z:x\leq z