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In mathematics, a '''locally finite poset''' is a [[partially ordered set]] ''P'' such that for all ''x'',&nbsp;''y''&nbsp;&isin;&nbsp;''P'', the [[poset#Interval|interval]] [''x'',&nbsp;''y''] consists of [[finite set|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'',&nbsp;''y''] of ''P'' a real number ''ƒ''(''x'',&nbsp;''y''). These functions form an [[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 [[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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[[Category:Order theory]]


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