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 Dushnik–Miller dimension of the partial order. first studied order dimension; for a more detailed treatment of this subject than provided here, see . The dimension of a poset P is the least integer t for which there exists a family of linear extensions of P so that, for every x and y in P, x precedes y in P if and only if it precedes y in all of the linear extensions. That is, An alternative definition of order dimension is the minimal number of total orders such that P embeds into their product with componentwise ordering i.e. if and only if for all i (, ). A family of linear orders on X is called a realizer of a poset P = (X,

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