Product order

In mathematics, given a partial order

⪯

⊑

on a set

, 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

declare that

Another possible ordering on

is the lexicographical order.

It is a total ordering if both

are totally ordered.

However the product order of two total orders is not in general total; for example, the pairs

are incomparable in the product order of the ordering

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

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.

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