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Product order

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In mathematics, given partial orders ⪯ {\displaystyle \preceq } and ⊑ {\displaystyle \sqsubseteq } on sets A {\displaystyle A} and B {\displaystyle B} , respectively, the product order (also called the coordinatewise order or componentwise order) is a partial order ≤ {\displaystyle \leq } on the Cartesian product A × B . {\displaystyle A\times B.} Given two pairs ( a 1 , b 1 ) {\displaystyle \left(a_{1},b_{1}\right)} and ( a 2 , b 2 ) {\displaystyle \left(a_{2},b_{2}\right)} in A × B , {\displaystyle A\times B,} declare that ( a 1 , b 1 ) ≤ ( a 2 , b 2 ) {\displaystyle \left(a_{1},b_{1}\right)\leq \left(a_{2},b_{2}\right)} if a 1 ⪯ a 2 {\displaystyle a_{1}\preceq a_{2}} and b 1 ⊑ b 2 . {\displaystyle b_{1}\sqsubseteq b_{2}.}