Order polytope

In mathematics, the order polytope of a finite <a href="/facts/Partially_ordered_set/qb4a3FAX">partially ordered set</a> is a <a href="/facts/Convex_polytope/2pRHcRgC">convex polytope</a> defined from the set. The points of the order polytope are the <a href="/facts/Monotonic_function/MjQK0YL2">monotonic functions</a> from the given set to the <a href="/facts/Unit_interval/W4dP8IJo">unit interval</a>, its vertices correspond to the <a href="/facts/Upper_set/hSyGJ2WB">upper sets</a> of the partial order, and its dimension is the number of elements in the partial order. The order polytope is a <a href="/facts/Distributive_polytope/BdcCDtsr">distributive polytope</a>, meaning that coordinatewise minima and maxima of pairs of its points remain within the polytope.
The order polytope of a partial order should be distinguished from the linear ordering polytope, a polytope defined from a number 
 
 
 
 n
 
 
 {\displaystyle n}
 
 as the <a href="/facts/Convex_hull/D4Id3JDS">convex hull</a> of <a href="/facts/Indicator_vector/j4nrF9T3">indicator vectors</a> of the sets of edges of 
 
 
 
 n
 
 
 {\displaystyle n}
 
-vertex <a href="/facts/Transitive_tournament/SVj5jWWE">transitive tournaments</a>.

Order polytope open-in-new

Order polytope