componentwise order

In mathematics, given two preordered sets $A$ and $B,$ the product order (also called the coordinatewise orderDavey & Priestley, ''Introduction to Lattices and Order'' (Second Edition), 2002, p. 18 or componentwise order) is a partial ordering on the Cartesian product $A \times B.$ Given two pairs $\left(a_1, b_1\right)$ and $\left(a_2, b_2\right)$ in $A \times B,$ declare that $\left(a_1, b_1\right) \leq \left(a_2, b_2\right)$ if and only if $a_1 \leq a_2$ and $b_1 \leq b_2.$

Another possible ordering on $A \times B$ 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 $(0, 1)$ and $(1, 0)$ are incomparable in the product order of the ordering $0 < 1$ 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 $A \neq \varnothing$ is a set and for every $a \in A,$ $\left(I_a, \leq\right)$ is a preordered set.
Then the on $\prod_ I_a$ is defined by declaring for any $i_ = \left(i_a\right)_$ and $j_ = \left(j_a\right)_$ in $\prod_ I_a,$ that

:$i_ \leq j_$ if and only if $i_a \leq j_a$ for every $a \in A.$

If every $\left(I_a, \leq\right)$ is a partial order then so is the product preorder.

Furthermore, given a set $A,$ the product order over the Cartesian product $\prod_ \$ can be identified with the inclusion ordering of subsets of $A.$

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.

References

See also

* Direct product of binary relations
* Examples of partial orders
* Star product, a different way of combining partial orders
* Orders on the Cartesian product of totally ordered sets
* Ordinal sum of partial orders
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Order theory