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

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

Offset: 1

Author

Arman Shamsgovara, May 28 2022

Keywords

• nonn
• more

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

• 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.

Crossrefs

Related algebraic structures: A027851, A006966.

Programs

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.