Ordinal Multiplication

Let (A,<=) and (B,<=) be totally ordered sets. Let C=A×B be the Cartesian product and define order as follows. For any a_1,a_2 in A and b_1,b_2 in B,

1. If a_1<a_2, then (a_1,b_1)<(a_2,b_2),

2. If a_1=a_2, then (a_1,b_1) and (a_2,b_2) compare the same way as b_1,b_2 (i.e., lexicographical order)

(Ciesielski 1997, p. 48; Rubin 1967; Suppes 1972). However, Dauben (1990, p. 104) and Moore (1982, p. 40) define multiplication in the reverse order.

Like addition, multiplication is not commutative, but it is associative,


An inductive definition for ordinal multiplication states that for any ordinal number alpha,

 alpha*(successor to beta)=alpha*beta+alpha.

If beta is a limit ordinal, then alpha*beta is the least ordinal greater than any ordinal in the set {alpha*gamma:gamma<beta} (Suppes 1972, p. 212).

See also

Ordinal Addition, Ordinal Exponentiation, Ordinal Number, Successor

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Ciesielski, K. Set Theory for the Working Mathematician. Cambridge, England: Cambridge University Press, 1997.Dauben, J. W. Georg Cantor: His Mathematics and Philosophy of the Infinite. Princeton, NJ: Princeton University Press, 1990.Moore, G. H. Zermelo's Axiom of Choice: Its Origin, Development, and Influence. New York: Springer-Verlag, 1982.Rubin, J. E. Set Theory for the Mathematician. New York: Holden-Day, 1967.Suppes, P. Axiomatic Set Theory. New York: Dover, 1972.

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Ordinal Multiplication

Cite this as:

Weisstein, Eric W. "Ordinal Multiplication." From MathWorld--A Wolfram Web Resource.

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