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Product order
From Wikipedia, the free encyclopedia
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
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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 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
[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]