In mathematics, a locally finite poset is a partially ordered set P such that for all x, y ∈ P, the interval [x, y] consists of finitely many elements.

Given a locally finite poset P we can define its incidence algebra. Elements of the incidence algebra are functions ƒ that assign to each interval [x, y] of P a real number ƒ(x, y). These functions form an associative algebra with a product defined by

( f ∗ g ) ( x , y ) := ∑ x ≤ z ≤ y f ( x , z ) g ( z , y ) . {\displaystyle (f*g)(x,y):=\sum _{x\leq z\leq y}f(x,z)g(z,y).}

There is also a definition of incidence coalgebra.

In theoretical physics a locally finite poset is also called a causal set and has been used as a model for spacetime.

  • Stanley, Richard P. Enumerative Combinatorics, Volume I. Cambridge University Press, 1997. Pages 98, 113–116.