Product order

In mathematics, given a partial order and on a set and , respectively, the product order (also called the coordinatewise order or componentwise order) is a partial ordering on the Cartesian product Given two pairs and in declare that if and Another possible ordering on is the lexicographical order, which is a total ordering. However the product order of two total orders is not in general total; for example, the pairs and are incomparable in the product order of the ordering with itself. The lexicographic combination of two total orders is a linear extension of their product order, and thus the product order is a subrelation of the lexicographic order. The Cartesian product with the product order is the categorical product in the of partially ordered sets with monotone functions. The product order generalizes to arbitrary (possibly infinitary) Cartesian products. Suppose is a set and for every is a preordered set. Then the on is defined by declaring for any and in that if and only if for every If every is a partial order then so is the product preorder. Furthermore, given a set the product order over the Cartesian product can be identified with the inclusion ordering of subsets of The notion applies equally well to preorders. The product order is also the categorical product in a number of richer categories, including lattices and Boolean algebras.