cp's OEIS Frontend

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

%I A354493 #30 Jun 01 2025 16:19:07
%S A354493 1,2,12,129,1852,33391,729629,19174600,658343783
%N A354493 Number of quantales on n elements, up to isomorphism.
%C A354493 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.
%D A354493 P. Eklund, J. G. García, U. Höhle, and J. Kortelainen, (2018). Semigroups in complete lattices. In Developments in Mathematics (Vol. 54). Springer Cham.
%D A354493 K. I. Rosenthal, Quantales and their applications. Longman Scientific and Technical, 1990.
%D A354493 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.
%D A354493 Arman Shamsgovara and P. Eklund, A Catalogue of Finite Quantales, GLIOC Notes, December 2019.
%H A354493 W. McCune, <a href="https://www.cs.unm.edu/~mccune/prover9">Prover9 and Mace4</a>.
%H A354493 Arman Shamsgovara, <a href="https://doi.org/10.1007/978-3-031-28083-2_14">Enumerating, Cataloguing and Classifying All Quantales on up to Nine Elements</a>, 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.
%o A354493 (Mace4)
%o A354493 assign(max_models,-1).
%o A354493 assign(domain_size,4).
%o A354493 formulas(assumptions).
%o A354493 % Comment: This will find all quantales on 4 elements, fixing
%o A354493 % 0 as the bottom and 3 as the top. Elements will be numbered
%o A354493 % 0-3. Results *must* be run through the companion program
%o A354493 % isofilter that is included with the downloads for mace4,
%o A354493 % otherwise the output will contain isomorphic duplicates!
%o A354493 % By changing the domain size, this file should be sufficient
%o A354493 % for up to 6 elements, but will crash for higher numbers.
%o A354493 (x*y)*z = x*(y*z).
%o A354493 (x v y) v z = x v (y v z).
%o A354493 x v y = y v x.
%o A354493 x v x = x.
%o A354493 x*(y v z) = (x*y) v (x*z).
%o A354493 (x v y)*z = (x*z) v (y*z).
%o A354493 0*x = 0.
%o A354493 x*0 = 0.
%o A354493 0 v x = x.
%o A354493 3 v x = 3.
%o A354493 end_of_list.
%o A354493 formulas(goals).
%o A354493 end_of_list.
%Y A354493 Related algebraic structures: A027851, A006966.
%K A354493 nonn,more
%O A354493 1,2
%A A354493 _Arman Shamsgovara_, May 28 2022