A027851 -id:A027851 - cp's OEIS frontend

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.

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.

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.

Related algebraic structures: A027851, A006966.

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.

A079175 Number of isomorphism classes of associative closed binary operations (semigroups) on a set of order n, listed by class size.

Original entry on oeis.org

1, 1, 2, 3, 2, 0, 7, 15, 2, 0, 0, 7, 5, 0, 62, 112, 2, 0, 0, 0, 6, 0, 0, 8, 0, 2, 51, 0, 47, 2, 576, 1221, 2, 0, 0, 0, 0, 6, 0, 0, 0, 0, 8, 0, 0, 4, 0, 48, 0, 0, 0, 0, 92, 0, 0, 42, 506, 0, 813, 32, 7397, 19684, 2, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8

Offset: 0

Christian van den Bosch (cjb(AT)cjb.ie), Jan 03 2003

Elements per row: 1,1,2,4,8,16,30,... (given by A027423, number of positive divisors of n!)

			Triangle T(n,k) begins:
  1;
  1;
  2, 3;
  2, 0, 7, 15;
  2, 0, 0, 7, 5, 0, 62, 112;
  2, 0, 0, 0, 6, 0, 0, 8, 0, 2, 51, 0, 47, 2, 576, 1221;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..217 (rows 0..8; row 8 was derived from data given in the Distler-Kelsey reference)
C. van den Bosch, Closed binary operations on small sets
A. Distler and T. Kelsey, The semigroups of order 9 and their automorphism groups, arXiv preprint arXiv:1301.6023 [math.CO], 2013.
Index entries for sequences related to semigroups

Row sums give A027851.

Cf. A023814, A027423 (row lengths), A079171, A079174, A079210.

A079174(n,k) + T(n,k) = A079171(n,k).
T(n, A027423(n)) = A058104(n).
A023814(n) = Sum_{k>=1} T(n,k)*A079210(n,k).

a(0)=1 prepended and terms a(16) and beyond from Andrew Howroyd, Jan 26 2022

A058112 Number of isomorphism classes of idempotent semigroups of order n.

Original entry on oeis.org

1, 1, 3, 10, 46, 251, 1682, 13213, 119826, 1228712

Offset: 0

Christian G. Bower, Nov 09 2000

Main diagonal of A058108 and A058137.

Cf. A027851, A351730 (labeled).

Updated Feb 19 2001

a(8)-a(9) added by Andrew Howroyd, Feb 17 2022

A350873 a(n) is the number of non-isomorphic flexible magmas with n elements (i.e., number of magmas with n elements which satisfy the identity (xy)x = x(yx) for all x and y).

Original entry on oeis.org

1, 1, 6, 191, 72148

Offset: 0

Joel Brennan, Jan 20 2022

			There are 10 non-isomorphic magmas with 2 elements, and 6 of these are flexible magmas, so a(2) = 6.

Wikipedia, Flexible algebra

Cf. A001329 (magmas), A027851.

A058104 Number of asymmetric semigroups of order n.

Original entry on oeis.org

1, 1, 3, 15, 112, 1221, 19684, 1458882, 3667253972, 105923135799007

Offset: 0

Christian G. Bower, Nov 09 2000

A. Distler and T. Kelsey, The semigroups of order 9 and their automorphism groups, arXiv preprint arXiv:1301.6023 [math.CO], 2013.
Index entries for sequences related to semigroups

Cf. A027851, A058113, A079175.

a(n) = A079175(n, A027423(n)).

Updated Feb 19 2001

a(8)-a(9) added from the Distler-Kelsey reference by Andrew Howroyd, Jan 26 2022

A383885 Number of nonisomorphic 3-nilpotent semigroups of order n.

Original entry on oeis.org

0, 0, 1, 9, 118, 4671, 1199989, 3661522792, 105931872028455, 24834563582168716305, 53061406576514239124327751, 2017720196187069550262596208732035, 2756576827989210680367439732667802738773384, 73919858836708511517426763179873538289329852786253510, 29599937964452484359589007277447538854227891149791717673581110642

Offset: 1

Elijah Beregovsky, May 13 2025

nonn

A semigroup S is nilpotent if there exists a natural number r such that the set S^r of all products of r elements of S has size 1.
If r is the smallest such number, then S is said to have nilpotency degree r.
This sequence counts semigroups S that have an element e such that for all x,y,z in S x*y*z = e.
In 1976 Kleitman, Rothschild and Spencer gave an argument asserting that the proportion of 3-nilpotent semigroups, amongst all semigroups of order n, is asymptotically 1. Later opinion regards their argument as incomplete, and no satisfactory proof has been found.

H. Jürgensen, F. Migliorini, and J. Szép, Semigroups. Akadémiai Kiadó (Publishing House of the Hungarian Academy of Sciences), Budapest, 1991.

Andreas Distler and James D. Mitchell, The number of nilpotent semigroups of degree 3, arXiv:1201.3529 [math.CO], 2012.
Igor Dolinka, D. G. FitzGerald, and James D. Mitchell, Semirigidity and the enumeration of nilpotent semigroups of index three, arXiv:2411.00466 [math.CO], 2024.
Pierre A. Grillet, Counting Semigroups, Communications in Algebra, 43(2), 574-596, (2014).
D. J. Kleitman, B. R. Rothschild, and J. H. Spencer, The number of semigroups of order n, Proc. Amer. Math. Soc. 55 (1976), 227-232.
Index entries for sequences related to semigroups

Cf. A023814, A027851.

Cf. A383871, A383886.

a(n) = A383871(n)/n! * (1+o(1)). See Grillet paper in Links.

For exact formula see the Distler and Mitchell paper.

A118581 Number of nonisomorphic semigroups of order <= n.

Original entry on oeis.org

1, 2, 7, 31, 219, 2134, 30768, 1658440, 3685688857, 105981863625149

Offset: 0

Jonathan Vos Post, May 07 2006

Semigroup analog of A063756 Number of groups of order <= n. a semigroup is an algebraic structure consisting of a set S closed under an associative binary operation (and thus is an associative groupoid). Some sources require that a semigroup have an identity element (in which case semigroups are identical to monoids). Not all sources agree that S should be nonempty. This sequence assumes that a semigroup may be empty and need not have an identity.

			a(7) = 1658440 = 1 + 1 + 5 + 24 + 188 + 1915 + 28634 + 1627672.

Cf. A001329, A001423, A001426, A023814, A027851, A029851, A058108, A063756, A079173.

a(n) = Sum_{i=0..n} A027851(i). a(n) = Sum_{i=0..n} (2*A001423(i) - A029851(i)).

a(8)-a(9) (using A027851) from Giovanni Resta, Jun 16 2016

A118601 Partial sums of A058129.

Original entry on oeis.org

1, 3, 10, 45, 273, 2510, 34069, 1703066

Offset: 1

Jonathan Vos Post, May 08 2006

Cf. A001329, A001423, A001426, A023814, A027851, A029851, A058108, A058132, A058133, A063756, A079173, A118581.

a(n) = SUM[i=1..n] A058129(i). a(n) = SUM[i=1..n] (2*A058133(i) - A058132(i)).

One more term from Jonathan Vos Post, Jul 20 2009

Edited by N. J. A. Sloane, Jul 25 2009

cp's OEIS Frontend

A079241 Number of isomorphism classes of associative non-commutative non-anti-associative non-anti-commutative closed binary operations on a set of order n.

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A029851 Number of self-converse semigroups of order n.

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

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Mace4

A079175 Number of isomorphism classes of associative closed binary operations (semigroups) on a set of order n, listed by class size.

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A058112 Number of isomorphism classes of idempotent semigroups of order n.

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A350873 a(n) is the number of non-isomorphic flexible magmas with n elements (i.e., number of magmas with n elements which satisfy the identity (xy)x = x(yx) for all x and y).

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A058104 Number of asymmetric semigroups of order n.

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A383885 Number of nonisomorphic 3-nilpotent semigroups of order n.

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A118581 Number of nonisomorphic semigroups of order <= n.

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A118601 Partial sums of A058129.

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