Locally finite poset

In mathematics, a locally finite poset is a <a href="/facts/Partially_ordered_set/qb4a3FAX">partially ordered set</a> P such that for all x, y ∈ P, the <a href="/facts/Poset/qb4a3FAX">interval</a> [x, y] consists of <a href="/facts/Finite_set/za1newlP">finitely</a> many elements.
Given a locally finite poset P we can define its <a href="/facts/Incidence_algebra/xrLTmg99">incidence algebra</a>. 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 <a href="/facts/Associative_algebra/DRfoF6KV">associative algebra</a> 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 <a href="/facts/Incidence_coalgebra/pvx52xS6">incidence coalgebra</a>.
In <a href="/facts/Theoretical_physics/Cs2Y12rQ">theoretical physics</a> a locally finite poset is also called a <a href="/facts/Causal_set/Le5SJx5Q">causal set</a> and has been used as a model for <a href="/facts/Spacetime/uRIN1x4b">spacetime</a>.

<ul><li><a href="/facts/Richard_P._Stanley/8bQ4QrLM">Stanley, Richard P.</a> Enumerative Combinatorics, Volume I. Cambridge University Press, 1997. Pages 98, 113–116.</li></ul>

Locally finite poset open-in-new

Locally finite poset