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Number of quantales on n elements, up to isomorphism.
7

%I #26 Mar 10 2023 11:03:58

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

%N Number of quantales on n elements, up to isomorphism.

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

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

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

%H W. McCune, <a href="https://www.cs.unm.edu/~mccune/prover9">Prover9 and Mace4</a>.

%H 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 (mace4)

%o assign(max_models,-1).

%o assign(domain_size,4).

%o formulas(assumptions).

%o % Comment: This will find all quantales on 4 elements, fixing

%o % 0 as the bottom and 3 as the top. Elements will be numbered

%o % 0-3. Results *must* be run through the companion program

%o % isofilter that is included with the downloads for mace4,

%o % otherwise the output will contain isomorphic duplicates!

%o % By changing the domain size, this file should be sufficient

%o % for up to 6 elements, but will crash for higher numbers.

%o (x*y)*z = x*(y*z).

%o (x v y) v z = x v (y v z).

%o x v y = y v x.

%o x v x = x.

%o x*(y v z) = (x*y) v (x*z).

%o (x v y)*z = (x*z) v (y*z).

%o 0*x = 0.

%o x*0 = 0.

%o 0 v x = x.

%o 3 v x = 3.

%o end_of_list.

%o formulas(goals).

%o end_of_list.

%Y Related algebraic structures: A027851, A006966.

%K nonn,more

%O 1,2

%A _Arman Shamsgovara_, May 28 2022