Dimension Theory for Ordered Sets

In 1930, E. Szpilrajn proved that any order relation on a set X can be extended to a linear order on X. It also follows that any order relation is the intersection of its linear extensions. B. Dushnik and E.W. Miller later defined the dimension of an ordered set P = 〈X;≤〉 to be the minimum number of linear extensions whose intersection is the ordering ≤.

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