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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,

2*omega=omega!=omega*2.
(1)

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

alpha*0=0
(2)
alpha*(successor to beta)=alpha*beta+alpha.
(3)

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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References

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. https://mathworld.wolfram.com/OrdinalMultiplication.html

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