OFFSET
1,2
COMMENTS
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.
LINKS
Stanley Burris and Simon Lee, Small models of the high school identities, International Journal of Algebra and Computation 2:2 (1992), pp. 139-178.
Stanley Burris and Simon Lee, Tarski's high school identities, Amer. Math. Monthly 100 (1993), 231-236.
Choiwah Chow, Mikoláš Janota, and João Araújo, Cube-based Isomorph-free Finite Model Finding, IOS ebook, Volume 392: ECAI 2024, Frontiers in Artificial Intelligence and Applications. See p. 4105.
FORMULA
Trivial upper bound: a(n) <= n^(3n^2+1). - Charles R Greathouse IV, Jun 19 2013
EXAMPLE
From Bert Dobbelaere, Sep 13 2020: (Start)
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).
CROSSREFS
KEYWORD
nonn,nice,hard,more
AUTHOR
Charles R Greathouse IV, Aug 21 2012
EXTENSIONS
a(4) from Bert Dobbelaere, Sep 13 2020
a(5)-a(6) from Choiwah Chow, Oct 21 2024
STATUS
approved