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A214396
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Number of HSI-algebras on n elements, up to isomorphism.
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0
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OFFSET
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1,2
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COMMENTS
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An HSI-algebra is a structure (1, +, *, ^) over some set such that Tarski's high-school identities hold: addition and multiplication are commutative and associative, multiplication distributes over addition, 1 is the multiplicative identity, x^1 = x, 1^x = 1, x^y * x^z = x^(y+z), (xy)^z = x^z * y^z, and (x^y)^z = x^(y*z).
Burris & Lee (1992) find a(3) = 44.
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LINKS
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FORMULA
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EXAMPLE
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The following operator definitions over the set of elements {1,A,B} is consistent with the identities. There are 44 such solutions that cannot be transformed into eachother by swapping symbols, hence a(3) = 44.
x + y | y = 1 A B x * y | y = 1 A B x ^ y | y = 1 A B
------+-------------- -------+-------------- -------+--------------
x = 1 | A A 1 x = 1 | 1 A B x = 1 | 1 1 1
A | A A A A | A A B A | A A 1
B | 1 A B B | B B B B | B B B
(End).
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CROSSREFS
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KEYWORD
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nonn,nice,hard,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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