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. Dushnik & Miller (1941) first studied order dimension; for a more detailed treatment of this subject than provided here, see Trotter (1992).
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 each of the linear extensions. That is,
An alternative definition of order dimension is as the minimal number of total orders such that P embeds to the product of these total orders for the componentwise ordering, in which if and only if for all i (Hiraguti 1955, Milner & Pouzet 1990).
...
Wikipedia