In mathematics, given a partial order and on a set and , respectively, the product order [1] [2] [3] [4] (also called the coordinatewise order [5] [3] [6] or componentwise order [2] [7]) 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. It is a total ordering if both and are totally ordered. 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. [3]
The Cartesian product with the product order is the categorical product in the category of partially ordered sets with monotone functions. [7]
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 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 [4]
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. [7]
In mathematics, given a partial order and on a set and , respectively, the product order [1] [2] [3] [4] (also called the coordinatewise order [5] [3] [6] or componentwise order [2] [7]) 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. It is a total ordering if both and are totally ordered. 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. [3]
The Cartesian product with the product order is the categorical product in the category of partially ordered sets with monotone functions. [7]
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 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 [4]
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. [7]