Rank
المؤلف:
Biggs, N. L.
المصدر:
Algebraic Graph Theory, 2nd ed. Cambridge, England: Cambridge University Press
الجزء والصفحة:
...
12-1-2022
1275
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 
is defined as 
, where 
is the number of vertices on 
and 
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 ![<span style=]()
{}" src="https://mathworld.wolfram.com/images/equations/Rank/Inline6.svg" style="height:22px; width:12px" /> has rank 0 (since it has no members and 0 is the least ordinal number), ![<span style=]()
{{}}" src="https://mathworld.wolfram.com/images/equations/Rank/Inline7.svg" style="height:22px; width:24px" /> has rank 1 (since ![<span style=]()
{}" src="https://mathworld.wolfram.com/images/equations/Rank/Inline8.svg" style="height:22px; width:12px" />, its only member, has rank 0), ![<span style=]()
{{{}}}" src="https://mathworld.wolfram.com/images/equations/Rank/Inline9.svg" style="height:22px; width:36px" /> has rank 2, and ![<span style=]()
{{},{{}},{{{}}},...}" src="https://mathworld.wolfram.com/images/equations/Rank/Inline10.svg" style="height:22px; width:135px" /> has rank 
. Every ordinal number has itself as its rank.
Mirimanoff (1917) showed that, assuming the class of urelements is a set, for any ordinal number 
, the class of all sets having rank 
is a set, i.e., not a proper class (Rubin 1967, p. 216) The number of sets having rank 
for 
, 1, ... are 1, 1, 2, 12, 65520, ... (OEIS A038081), and the number of sets having rank at most 
is 
, 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 
.
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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