**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

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