## Order Dimension

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 **DushnikâMiller 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 ...

**Order Dimension**- K-dimension and 2-dimension

... 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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