A214396 Number of HSI-algebras on n elements, up to isomorphism.

1, 5, 44, 657, 13577, 672740

Offset: 1

Author

Charles R Greathouse IV, Aug 21 2012

Keywords

• nonn
• nice
• hard
• more

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.

Examples

			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).
		

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.

Crossrefs

Cf. A007459.

Formula

Trivial upper bound: a(n) <= n^(3n^2+1). - Charles R Greathouse IV, Jun 19 2013

Extensions

a(4) from Bert Dobbelaere, Sep 13 2020

a(5)-a(6) from Choiwah Chow, Oct 21 2024