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%I #26 Sep 13 2020 14:44:15
%S 1,5,44,657
%N Number of HSI-algebras on n elements, up to isomorphism.
%C 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).
%C Burris & Lee (1992) find a(3) = 44.
%H S. Burris and S. Lee, <a href="https://doi.org/10.1142/S0218196792000104">Small models of the high school identities</a>, International Journal of Algebra and Computation 2:2 (1992), pp. 139-178.
%H S. Burris and S. Lee, <a href="http://www.jstor.org/stable/2324454">Tarski's high school identities</a>, Amer. Math. Monthly 100 (1993), 231-236.
%F Trivial upper bound: a(n) <= n^(3n^2+1). - _Charles R Greathouse IV_, Jun 19 2013
%e From _Bert Dobbelaere_, Sep 13 2020: (Start)
%e 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.
%e x + y | y = 1 A B x * y | y = 1 A B x ^ y | y = 1 A B
%e ------+-------------- -------+-------------- -------+--------------
%e x = 1 | A A 1 x = 1 | 1 A B x = 1 | 1 1 1
%e A | A A A A | A A B A | A A 1
%e B | 1 A B B | B B B B | B B B
%e (End).
%Y Cf. A007459.
%K nonn,nice,hard,more
%O 1,2
%A _Charles R Greathouse IV_, Aug 21 2012
%E a(4) from _Bert Dobbelaere_, Sep 13 2020
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