View source for Locally finite poset

In mathematics, a '''locally finite poset''' is a [[Partially ordered set|partially ordered set]] ''P'' such that for all ''x'',&nbsp;''y''&nbsp;&isin;&nbsp;''P'', the interval [''x'',&nbsp;''y''] consists of [[Finite set|finitely]] many elements.

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

: <math>(f * g)(x,y):=\sum_{x \leq z \leq y} f(x,z) g(z,y).</math>

There is also a definition of ''incidence coalgebra''.

In [[Physics:Theoretical physics|theoretical physics]] a locally finite poset is also called a causal set and has been used as a model for [[Physics:Spacetime|spacetime]].

==References==
[[Biography:Richard P. Stanley|Stanley, Richard P.]] Enumerative Combinatorics, Volume I. Cambridge University Press, 1997. Pages 98, 113–116.

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[[Category:Order theory]]



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