I belabor the Cartesian product here in more detail than most of these pages because it is important to so much of mathematics. René Descartes made his seminal contribution to geometry and algebra by building a bridge between them. He invented Cartesian coordinates on the plane to identify each point with an ordered pair of numbers <x, y>. He could thus relate the algebraic equation x2 + y2 = 1 to a circle in the plane. Perhaps this was also the invention of the ordered pair or ordered n-tuple.

Given two sets, A and B, their Cartesian product, A×B (or A╳B depending on your taste and browser), is the set of ordered pairs, <a, b> where a∊A and b∊B.
A╳B = {<a, b> | a∊A & b∊B}

The direct sum is a closely related concept in abstract algebra. Given two algebras A and B, isomorphic or not, one can form A⊕B which is a new algebra whose elements are ordered pairs <a, b> with a∊A and b∊B. If the two algebras share a binary operator + then <a, b> + <c, d> is automatically taken to mean <a+c, b+d> in the new algebra. Common axioms from A and B, in so far as they involve only universal quantifiers are inherited by A⊕B. Other axioms such as the group axiom ∀a∀b∃c ac=b are also inherited.

With normal axioms from set theory the existence of models for axioms for some algebra provide for the existence of models for the cartesian product via the mechanism of ordered pairs and indeed such resulting models are isomorphic.

This works for groups and rings but not for fields whose axiom ∀x((∃y∀z (xy)z = z) ∨ ∀z(x+z = z)) lacks the logical properties to be inherited.

The generalization to Cartesian products of more than two algebras can be done as (A⊕B)⊕C whose elements are <<a, b>, c>, or as ordered triples <a, b, c>. There is scarcely a difference here and these two perspectives are equivalent.


Cartesian products of topologies are likewise very important. If X and Y are two topologies and Bx and By are bases of them respectively, then {x×y | x∊Bx & y∊By} form a base for X×Y. The cartesian product of two line segments, taken as topologies, is a square. The cartesian product of two circles, is a torus.