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| In [[mathematics]], '''Laver tables''' (named after [[Richard Laver]], who discovered them towards the end of the 1980s in connection with his works on [[set theory]]) are tables of numbers that have certain properties.
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| == Definition ==
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| For a given a [[natural number]] ''n'', one can define the ''n''-th Laver table (with 2<sup>''n''</sup> rows and columns) by setting
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| :<math>L_n(p, q) := p \star q</math>,
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| where ''p'' denotes the row and ''q'' denotes the column of the entry. Define
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| :<math>p \star 1 := p + 1 \mod 2^n</math> | |
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| and then calculate the remaining entries of each row from the ''m''-th to the first using the equation
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| :<math>p \star (q \star r) := (p \star q) \star (p \star r)</math>
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| The resulting table is then called the ''n''-th Laver table; for example, for ''n'' = 2, we have:
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| There is no known [[closed-form expression]] to calculate the entries of a Laver table directly, and it is in fact suspected that such a formula does not exist.{{citation needed|date=August 2010}}
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| == Periodicity == | |
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| When looking at the first row of entries in a Laver table, it can be seen that the entries repeat with a certain periodicity ''m''. This periodicity is always a power of 2; the first few periodicities are 1, 1, 2, 4, 4, 8, 8, 8, 8, 16, 16, ... {{OEIS|A098820}}. The sequence is increasing, and it was proved in 1995 by Richard Laver that under the assumption that there exists a [[rank-into-rank]] (a [[large cardinal]]), it actually increases without bound. Nevertheless, it grows extremely slowly; Randall Dougherty showed that the first ''n'' for which the table entries' period can possibly be 32 is A(9,A(8,A(8,255))), where A denotes the [[Ackermann function]].
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| == References ==
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| * Patrick Dehornoy, "''Das Unendliche als Quelle der Erkenntnis''", in: Spektrum der Wissenschaft Spezial 1/2001, pp. 86–90
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| == Further reading ==
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| * R. Laver, ''On the Algebra of Elementary Embeddings of a Rank into Itself'', Advances in Mathematics 110, p. 334, 1995 {{arXiv|math.LO/9204204}}
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| * R. Dougherty, ''Critical Points in an Algebra of Elementary Embeddings'', Annals of Pure and Applied Logic 65, p. 211, 1993 {{arXiv|math.LO/9205202}}
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| * Patrick Dehornoy, ''Diagrams colourings and applications'', Proceedings of the East Asian School of Knots, Links and Related Topics, 2004 ([http://knot.kaist.ac.kr/2004/proceedings/DEHORNOY.pdf online])
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| {{DEFAULTSORT:Laver Table}}
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| [[Category:Mathematical logic]]
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| [[Category:Combinatorics]]
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