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Order Isomorphic

Two totally ordered sets (A,<=) and (B,<=) are order isomorphic iff there is a bijection f from A to B such that for all a_1,a_2 in A,

a_1<=a_2 iff f(a_1)<=f(a_2)

(Ciesielski 1997, p. 38). In other words, A and B are equipollent ("the same size") and there is an order preserving mapping between the two.

Dauben (1990) and Suppes (1972) call this property "similar." The definition works equally well on partially ordered sets.

See also

Avoided Pattern, Contained Pattern, Partially Ordered Set, Permutation Pattern, Totally Ordered Set

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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.Mansour, T. "Permutations Avoiding a Pattern from S_k and at Least Two Patterns from S_3." 31 Jul 2000. https://arxiv.org/abs/math/0007194.Suppes, P. Axiomatic Set Theory. New York: Dover, 1972.

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Order Isomorphic

Cite this as:

Weisstein, Eric W. "Order Isomorphic." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/OrderIsomorphic.html

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