### Posets

**Definition **A *partially ordered set *or *poset *is a set equipped with a relation that is reflexive, antisymmetric, and transitive. That is, for all :

- .
- If and , then (antisymmetry)
- If and , then (transitivity).

We say that is *covered *by , written , if and there is no such that . Two posets are *isomorphic * if there is a bijection that is order-persverin; that is, in iff in .

We’ll usually assume that is finite.

### Ranked posets

**Definition **A chain is *saturated *if it is not properly contained in any other chain from to ; equivalently, if for every . In this case, the number is the *length *of the chain. A poset is *ranked *if for every , all saturated chains with top element have the same length; this number is called the *rank * of and denoted . A poset is *graded *if it is ranked and bounded.

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

### Simplicial complexes

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

### The incidence algebra of a poset

Let be a poset and let denote the set of (nonempty) intervals of , i.e., the sets

for all .

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

**Definition (Incidence algebra) **The *incidence algebra *is the set of funciton (“incidence funcitons”), made into a -vector space with pointwise addition, subtraction and scalar multiplication, and equipped with the *convolution product *

It is often convenient to set if . 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 has a left/right two side side-convolution inverse if and only if for all . In that case, the inverse is given by the formula