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Rank


The word "rank" refers to several related concepts in mathematics involving graphs, groups, matrices, quadratic forms, sequences, set theory, statistics, and tensors.

In graph theory, the graph rank of a graph G is defined as r(G)=n-c, where n is the number of vertices on G and c is the number of connected components (Biggs 1993, p. 25).

In set theory, rank is a (class) function from sets to ordinal numbers. The rank of a set is the least ordinal number greater than the rank of any member of the set (Mirimanoff 1917; Moore 1982, pp. 261-262; Rubin 1967, p. 214). The proof that rank is well-defined uses the axiom of foundation.

For example, the empty set {} has rank 0 (since it has no members and 0 is the least ordinal number), {{}} has rank 1 (since {}, its only member, has rank 0), {{{}}} has rank 2, and {{},{{}},{{{}}},...} has rank omega. Every ordinal number has itself as its rank.

Mirimanoff (1917) showed that, assuming the class of urelements is a set, for any ordinal number alpha, the class of all sets having rank alpha is a set, i.e., not a proper class (Rubin 1967, p. 216) The number of sets having rank k for k=0, 1, ... are 1, 1, 2, 12, 65520, ... (OEIS A038081), and the number of sets having rank at most k is 2^(2^(·^(·^(·^2))))_()_(k), 1, 2, 4, 16, 65536, ... (OEIS A014221).

The rank of a mathematical object is defined whenever that object is free. In general, the rank of a free object is the cardinal number of the free generating subset G.


See also

Bundle Rank, Graph Rank, Group Rank, Lie Algebra Rank, Matrix Rank, Ordinal Number, Quadratic Form Rank, Rank-Nullity Theorem, Rank Order Correlation Coefficient, Sequence Rank, Statistical Rank, Tensor Rank

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References

Biggs, N. L. Algebraic Graph Theory, 2nd ed. Cambridge, England: Cambridge University Press, p. 73, 1993.Mirimanoff, D. "Les antinomies de Russell et de Burali-Forti et le problème fondamental de la théorie des ensembles." Enseign. math. 19, 37-52, 1917.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.Sloane, N. J. A. Sequences A014221 and A038081 in "The On-Line Encyclopedia of Integer Sequences."

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Rank

Cite this as:

Weisstein, Eric W. "Rank." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Rank.html

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