A354493 Number of quantales on n elements, up to isomorphism.

1, 2, 12, 129, 1852, 33391, 729629, 19174600, 658343783

Offset

1,2

Comments

A quantale is an algebraic structure (X,*,v) composed of a set X of elements, a semigroup operator "*" and a supremum operator "v" (in the sense of lattices) such that * distributes over v: x * (y v z) = (x * y) v (x * z) and (x v y) * z = (x * z) v (y * z) for all elements x,y,z in X. In addition the bottom element corresponding to v, denoted 0, must satisfy x * 0 = 0 * x = 0.

References

P. Eklund, J. G. García, U. Höhle, and J. Kortelainen, (2018). Semigroups in complete lattices. In Developments in Mathematics (Vol. 54). Springer Cham.

K. I. Rosenthal, Quantales and their applications. Longman Scientific and Technical, 1990.

Arman Shamsgovara, A catalogue of every quantale of order up to 9 (abstract, to appear), LINZ2022, 39th Linz Seminar on Fuzzy Set Theory, Linz, Austria.

Arman Shamsgovara and P. Eklund, A Catalogue of Finite Quantales, GLIOC Notes, December 2019.

Links

Table of n, a(n) for n=1..9.

W. McCune, Prover9 and Mace4.

Arman Shamsgovara, Enumerating, Cataloguing and Classifying All Quantales on up to Nine Elements, In: Glück, R., Santocanale, L., and Winter, M. (eds), Relational and Algebraic Methods in Computer Science (RAMiCS 2023) Lecture Notes in Computer Science, Springer, Cham, Vol. 13896.

Prog

(mace4)
assign(max_models, -1).
assign(domain_size, 4).
formulas(assumptions).
% Comment: This will find all quantales on 4 elements, fixing
% 0 as the bottom and 3 as the top. Elements will be numbered
% 0-3. Results *must* be run through the companion program
% isofilter that is included with the downloads for mace4,
% otherwise the output will contain isomorphic duplicates!
% By changing the domain size, this file should be sufficient
% for up to 6 elements, but will crash for higher numbers.
(x*y)*z = x*(y*z).
(x v y) v z = x v (y v z).
x v y = y v x.
x v x = x.
x*(y v z) = (x*y) v (x*z).
(x v y)*z = (x*z) v (y*z).
0*x = 0.
x*0 = 0.
0 v x = x.
3 v x = 3.
end_of_list.
formulas(goals).
end_of_list.

Crossrefs

Related algebraic structures: A027851, A006966.

Sequence in context: A014235 A098628 A123553 * A079199 A303926 A214759

Adjacent sequences: A354490 A354491 A354492 * A354494 A354495 A354496

Keyword

nonn,more

Author

Arman Shamsgovara, May 28 2022

Status

approved