In mathematics, the dimension of a partially ordered set (poset) is the smallest number of total orders the intersection of which gives rise to the partial order. This concept is also sometimes called the order dimension or the Dushnik–Miller dimension of the partial order. Dushnik & Miller (1941) first studied order dimension; for a more detailed treatment of this subject than provided here, see Trotter (1992).
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Some articles on order dimension:
... the incidence poset of a graph G has order dimension two if and only if the graph is a path or a subgraph of a path ... possible realizer for the incidence poset consists of two total orders that (when restricted to the graph's vertices) are the reverse of each other otherwise, the intersection of the two orders ... But two total orders on the vertices that are the reverse of each other can realize any subgraph of a path, by including the edges of the path in the ordering ...
... A generalization of dimension is the notion of k-dimension (written ) which is the minimal number of chains of length at most k in whose product the partial order can be embedded ... In particular, the 2-dimension of an order can be seen as the size of the smallest set such that the order embeds in the containment order on this set ...
... has been generalized by Brightwell and Trotter (1993, 1997) to a tight bound on the dimension of the height-three partially ordered sets formed analogously from the vertices, edges and ... convex polytopes, as there exist four-dimensional polytopes whose face lattices have unbounded order dimension ... for abstract simplicial complexes, the order dimension of the face poset of the complex is at most 1 + d, where d is the minimum dimension of a Euclidean space in ...
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