Realizers
A family of linear orders on X is called a realizer of a poset P = (X, <P) if
- ,
which is to say that for any x and y in X, x <P y precisely when x <1 y, x <2 y, ..., and x <t y. Thus, an equivalent definition of the dimension of a poset P is "the least cardinality of a realizer of P."
It can be shown that any nonempty family R of linear extensions is a realizer of a finite partially ordered set P if and only if, for every critical pair (x,y) of P, y <i x for some order <i in R.
Read more about this topic: Order Dimension