Product order

In mathematics, given a partial order $\preceq$ and $\sqsubseteq$ on a set $A$ 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 ordering $\leq$ 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 $a_{1}\preceq a_{2}$ and $b_{1}\sqsubseteq b_{2}.$

Another possible ordering on $A\times B$ is the lexicographical order. It is a total ordering 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 ordering $0<1$ 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 $A\neq \varnothing$ is a set and for every $a\in A,$ $\left(I_{a},\leq \right)$ 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 }\leq j_{\bullet }

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 $\prod _{a\in A}\{0,1\}$ can be identified with the inclusion ordering 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]

References

^ Neggers, J.; Kim, Hee Sik (1998), "4.2 Product Order and Lexicographic Order", Basic Posets, World Scientific, pp. 64–78, ISBN 9789810235895
^ ^a ^b Sudhir R. Ghorpade; Balmohan V. Limaye (2010). A Course in Multivariable Calculus and Analysis. Springer. p. 5. ISBN 978-1-4419-1621-1.
^ ^a ^b ^c Egbert Harzheim (2006). Ordered Sets. Springer. pp. 86–88. ISBN 978-0-387-24222-4.
^ ^a ^b Victor W. Marek (2009). Introduction to Mathematics of Satisfiability. CRC Press. p. 17. ISBN 978-1-4398-0174-1.
^ Davey & Priestley, Introduction to Lattices and Order (Second Edition), 2002, p. 18
^ Alexander Shen; Nikolai Konstantinovich Vereshchagin (2002). Basic Set Theory. American Mathematical Soc. p. 43. ISBN 978-0-8218-2731-4.
^ ^a ^b ^c Paul Taylor (1999). Practical Foundations of Mathematics. Cambridge University Press. pp. 144–145 and 216. ISBN 978-0-521-63107-5.

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[1] Neggers, J.; Kim, Hee Sik (1998), "4.2 Product Order and Lexicographic Order", Basic Posets, World Scientific, pp. 64–78, ISBN 9789810235895

[anal-2] Sudhir R. Ghorpade; Balmohan V. Limaye (2010). A Course in Multivariable Calculus and Analysis. Springer. p. 5. ISBN 978-1-4419-1621-1.

[Harzheim-3] Egbert Harzheim (2006). Ordered Sets. Springer. pp. 86–88. ISBN 978-0-387-24222-4.

[Marek-4] Victor W. Marek (2009). Introduction to Mathematics of Satisfiability. CRC Press. p. 17. ISBN 978-1-4398-0174-1.

[5] Davey & Priestley, Introduction to Lattices and Order (Second Edition), 2002, p. 18

[6] Alexander Shen; Nikolai Konstantinovich Vereshchagin (2002). Basic Set Theory. American Mathematical Soc. p. 43. ISBN 978-0-8218-2731-4.

[taylor-7] Paul Taylor (1999). Practical Foundations of Mathematics. Cambridge University Press. pp. 144–145 and 216. ISBN 978-0-521-63107-5.

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v t e Order theory
Topics Glossary Category
Key concepts	Binary relation Boolean algebra Cyclic order Lattice Partial order Preorder Total order Weak ordering
Results	Boolean prime ideal theorem Cantor–Bernstein theorem Cantor's isomorphism theorem Dilworth's theorem Dushnik–Miller theorem Hausdorff maximal principle Knaster–Tarski theorem Kruskal's tree theorem Laver's theorem Mirsky's theorem Szpilrajn extension theorem Zorn's lemma
Properties & Types (list)	Antisymmetric Asymmetric Boolean algebra topics Completeness Connected Covering Dense Directed (Partial) Equivalence Foundational Heyting algebra Homogeneous Idempotent Lattice Bounded Complemented Complete Distributive Join and meet Reflexive Partial order Chain-complete Graded Eulerian Strict Prefix order Preorder Total Semilattice Semiorder Symmetric Total Tolerance Transitive Well-founded Well-quasi-ordering (Better) (Pre) Well-order
Constructions	Composition Converse/Transpose Lexicographic order Linear extension Product order Reflexive closure Series-parallel partial order Star product Symmetric closure Transitive closure
Topology & Orders	Alexandrov topology & Specialization preorder Ordered topological vector space Normal cone Order topology Order topology Topological vector lattice Banach Fréchet Locally convex Normed
Related	Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet Order morphism Embedding Isomorphism Order type Ordered field Positive cone of an ordered field Ordered vector space Partially ordered Positive cone of an ordered vector space Riesz space Partially ordered group Positive cone of a partially ordered group Upper set Young's lattice

See also

References