Product order

rdf:langString Product order

rdf:langString In mathematics, given two preordered sets and 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 only if and Another possible ordering on is the lexicographical order, which is a total ordering. However the product order of two totally ordered sets is not in general total; for example, the pairs and are incomparable in the product order of the ordering with itself. The lexicographic order of totally ordered sets 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 category 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 product preorder 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.