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.
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
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