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In mathematics, given partial orders $⪯$ and $⊑$ on sets $A$ and $B$, respectively, the product order^[1][2][3][4] (also called the coordinatewise order^[5][3][6] or componentwise order^[2][7]) is a partial order $\le$ on the Cartesian product $A\times B.$ Given two pairs $(a_1,b_1)$ and $(a_2,b_2)$ in $A\times B,$ declare that $(a_1,b_1)\le (a_2,b_2)$ if $a_1⪯a_2$ and $b_1⊑b_2.$

Another possible order on $A\times B$ is the lexicographical order. It is a total order if both $A$ and $B$ are totally ordered. However the product order of two total orders is not in general total; for example, the pairs $(0,1)$ and $(1,0)$ are incomparable in the product order of the order $0<1$ 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 $A\ne \varnothing$ is a set and for every $a\in A,$ $(I_a,\le )$ is a preordered set. Then the product preorder on $\prod _{a\in A}{I}_a$ is defined by declaring for any $i_{\bullet }={\left(i_a\right)}_{a\in A}$ and $j_{\bullet }={\left(j_a\right)}_{a\in A}$ in $\prod _{a\in A}{I}_a,$ that

$i_{\bullet }\le j_{\bullet }$ if and only if $i_a\le j_a$ for every $a\in A.$

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

Furthermore, given a set $A,$ the product order over the Cartesian product $\prod _{a\in A}\{0,1\}$ can be identified with the inclusion order of subsets of $A.$^[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]

• 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
• Ordered vector space – Vector space with a partial order