Locally finite poset

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):=\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.

References

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

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