In mathematics, the order polytope of a finite partially ordered set is a convex polytope defined from the set.
The points of the order polytope are the monotonic functions from the given set to the unit interval, its vertices correspond to the upper sets of the partial order, and its dimension is the number of elements in the partial order.
The order polytope is a distributive polytope, 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
as the convex hull of indicator vectors of the sets of edges of
[1] A partially ordered set is a pair
is a binary relation on pairs of elements of
In this case, the collection of all functions
to the real numbers forms a finite-dimensional vector space, with pointwise addition of functions as the vector sum operation.
The dimension of the space is just the number of elements of
The order polytope is defined to be the subset of this space consisting of functions
with the following two properties:[2][3] For example, for a partially ordered set consisting of two elements
For this example, the order polytope consists of all points in the
The vertices of the order polytope consist of monotonic functions from
[2] The facets of the order polytope are of three types:[2] The facets can be considered in a more symmetric way by introducing special elements
to 0 and 1 respectively, and keeping only inequalities of the third type for the resulting augmented partially ordered set.
, the faces of all dimensions of the order polytope correspond 1-to-1 with quotients of the partial order.
Each face is congruent to the order polytope of the corresponding quotient partial order.
Each point of the unit cube whose coordinates are all distinct lies in a unique one of these orthoschemes, the order simplex for the linear order of its coordinates.
[2][3] More generally, an order polytope can be partitioned into order simplices in a canonical way, with one simplex for each linear extension of the corresponding partially ordered set.
multiplied by the number of linear extensions of the corresponding partially ordered set.
[2][3] This connection between the number of linear extensions and volume can be used to approximate the number of linear extensions of any partial order efficiently (despite the fact that computing this number exactly is #P-complete) by applying a randomized polynomial-time approximation scheme for polytope volume.
[4] The Ehrhart polynomial of the order polytope is a polynomial whose values at integer values
give the number of integer points in a copy of the polytope scaled by a factor of
For the order polytope, the Ehrhart polynomial equals (after a minor change of variables) the order polynomial of the corresponding partially ordered set.
This polynomial encodes several pieces of information about the polytope including its volume (the leading coefficient of the polynomial and its number of vertices (the sum of coefficients).
[2][3] By Birkhoff's representation theorem for finite distributive lattices, the upper sets of any partially ordered set form a finite distributive lattice, and every finite distributive lattice can be represented in this way.
[5] The upper sets correspond to the vertices of the order polytope, so the mapping from upper sets to vertices provides a geometric representation of any finite distributive lattice.
Under this representation, the edges of the polytope connect comparable elements of the lattice.
give the order polytope the structure of a continuous distributive lattice, within which the finite distributive lattice of Birkhoff's theorem is embedded.