A029851 -id:A029851 - cp's OEIS frontend

A001423 Number of semigroups of order n, considered to be equivalent when they are isomorphic or anti-isomorphic (by reversal of the operator).

Original entry on oeis.org

1, 1, 4, 18, 126, 1160, 15973, 836021, 1843120128, 52989400714478, 12418001077381302684

Offset: 0

N. J. A. Sloane

David Nacin, "Puzzles, Parity Maps, and Plenty of Solutions", Chapter 15, The Mathematics of Various Entertaining Subjects: Volume 3 (2019), Jennifer Beineke & Jason Rosenhouse, eds. Princeton University Press, Princeton and Oxford, p. 245.
R. J. Plemmons, There are 15973 semigroups of order 6, Math. Algor., 2 (1967), 2-17; 3 (1968), 23.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

A. de Vries, Formal Languages: An Introduction
Andreas Distler, Classification and Enumeration of Finite Semigroups, A Thesis Submitted for the Degree of PhD, University of St Andrews (2010).
Andreas Distler and Tom Kelsey, The Monoids of Order Eight and Nine, in Intelligent Computer Mathematics, Lecture Notes in Computer Science, Volume 5144/2008, Springer-Verlag. [From _N. J. A. Sloane_, Jul 10 2009]
A. Distler and T. Kelsey, The semigroups of order 9 and their automorphism groups, arXiv preprint arXiv:1301.6023 [math.CO], 2013.
Andreas Distler, Chris Jefferson, Tom Kelsey, and Lars Kotthoff, The Semigroups of Order 10, in: M. Milano (Ed.), Principles and Practice of Constraint Programming, 18th International Conference, CP 2012, Québec City, QC, Canada, October 8-12, 2012, Proceedings (LNCS, volume 7514), pp. 883-899, Springer-Verlag Berlin Heidelberg 2012.
Remigiusz Durka and Kamil Grela, On the number of possible resonant algebras, arXiv:1911.12814 [hep-th], 2019.
G. E. Forsythe, SWAC computes 126 distinct semigroups of order 4, Proc. Amer. Math. Soc. 6, (1955). 443-447.
H. Juergensen and P. Wick, Die Halbgruppen von Ordnungen <= 7, Semigroup Forum, 14 (1977), 69-79.
H. Juergensen and P. Wick, Die Halbgruppen von Ordnungen <= 7, annotated and scanned copy.
Daniel J. Kleitman, Bruce L. Rothschild and Joel H. Spencer, The number of semigroups of order n, Proc. Amer. Math. Soc., 55 (1976), 227-232.
R. J. Plemmons, There are 15973 semigroups of order 6 (annotated and scanned copy)
Eric Postpischil Associativity Problem, Posting to sci.math newsgroup, May 21 1990.
S. Satoh, K. Yama, and M. Tokizawa, Semigroups of order 8, Semigroup Forum 49 (1994), 7-29.
N. J. A. Sloane, Overview of A001329, A001423-A001428, A258719, A258720.
T. Tamura, Some contributions of computation to semigroups and groupoids, pp. 229-261 of J. Leech, editor, Computational Problems in Abstract Algebra. Pergamon, Oxford, 1970. (Annotated and scanned copy)
Eric Weisstein's World of Mathematics, Semigroup.
Index entries for sequences related to semigroups

Cf. A001426, A023814, A058107, A058123, A151823.

a(n) = (A027851(n) + A029851(n))/2.

a(9) added by Andreas Distler, Jan 12 2011

a(10) from Distler et al. 2012, added by Andrey Zabolotskiy, Nov 08 2018

A027851 Number of nonisomorphic semigroups of order n.

Original entry on oeis.org

1, 1, 5, 24, 188, 1915, 28634, 1627672, 3684030417, 105978177936292

Offset: 0

Christian G. Bower, Dec 13 1997, updated Feb 19 2001

Özgur Akgun, Mun See Chang, Ian P. Gent, and Christopher Jefferson, Breaking the Symmetries of Indistinguishable Objects, arXiv:2503.17251 [cs.AI], 2025. See pp. 14, 17.
Peter Cameron's Blog, The combinatorial explosion, Posted 18/02/2016.
Andreas Distler, Classification and Enumeration of Finite Semigroups, A Thesis Submitted for the Degree of PhD, University of St Andrews (2010).
A. Distler and T. Kelsey, The semigroups of order 9 and their automorphism groups, arXiv preprint arXiv:1301.6023 [math.CO], 2013.
C. Noebauer, Home page
C. Noebauer, The Numbers of Small Rings
C. Noebauer, Thesis on the enumeration of near-rings
Eric Postpischil Posting to sci.math newsgroup, May 21 1990
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.
Jeremy G. Sumner, Michael D. Woodhams, Lie-Markov models derived from finite semigroups, arXiv:1709.00520 [math.GR], 2017.
Michael Torpey, Semigroup congruences: computational techniques and theoretical applications, Ph.D. Thesis, University of St. Andrews (Scotland, 2019).
A. de Vries, Formal Languages: An Introduction, 2012.
Eric Weisstein's World of Mathematics, Semigroup.
Index entries for sequences related to semigroups

Cf. A001426, A023814, A058108.

Cf. A001423, A029851, A079173, A001329.

a(n) = A001423(n)*2 - A029851(n).

a(n) + A079173(n) = A001329(n).

a(8)-a(9) from Andreas Distler, Jan 13 2011

A058118 Triangle: self-converse semigroups of order n with k idempotents.

Original entry on oeis.org

1, 2, 1, 5, 5, 2, 18, 24, 16, 6, 93, 123, 112, 58, 19, 930, 754, 764, 542, 254, 68, 22899, 6721, 5567, 4751, 2933, 1214, 285, 1937266, 123988, 49969, 41939, 31392, 17396, 6513, 1376, 613365656, 8265721, 739317, 410158, 328937, 223226, 113160, 38979, 7510

Offset: 1

Christian G. Bower, Nov 10 2000

			1; 2,1; 5,5,2; 18,24,16,6; 93,123,112,58,19; ...

Andreas Distler, Classification and Enumeration of Finite Semigroups, A Thesis Submitted for the Degree of PhD, University of St Andrews (2010).
Index entries for sequences related to semigroups

Row sums give A029851. Main diagonal: A058122. Columns 1-3: A058119-A058121.

More terms from Andreas Distler, Jan 13 2011

A058106 Number of self-converse asymmetric semigroups of order n.

Original entry on oeis.org

1, 1, 3, 9, 44, 271, 2246, 33672

Offset: 0

Christian G. Bower, Nov 09 2000

Index entries for sequences related to semigroups

Cf. A029851.

Updated Feb 19 2001

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

A186117 Number of nonisomorphic semigroups of order n minus number of groups of order n.

Original entry on oeis.org

0, 4, 23, 186, 1914, 28632, 1627671, 3684030412, 105978177936290

Offset: 1

Jonathan Vos Post, Feb 13 2011

In a sense, this measures the increase in combinatorial structures available by dropping the requirement of inverses, and an identity element, in moving from the group axioms to the semigroup axioms. A semigroup is mathematical object defined for a set and a binary operator in which the multiplication operation is associative. No other restrictions are placed on a semigroup; thus a semigroup need not have an identity element and its elements need not have inverses within the semigroup. Other sequences may be derived by considering commutative semigroups and commutative groups, self-converse semigroup, counting idempotents, and the like.

			a(1) = 0 because there are unique groups and semigroups of order 1, so 1 - 1  = 0.
a(2) = 4 because there are 5 semigroups of order 2 groups and a unique group of order 2, so 5 - 1  = 4.

Eric W. Weisstein, Finite Group
Eric W. Weisstein, Semigroup

Cf. A000001, A027851, A001423, A029851, A001426, A023814, A058108, A079173, A001329, A186116.

a(n) = A027851(n) - A000001(n).

cp's OEIS Frontend

A001423 Number of semigroups of order n, considered to be equivalent when they are isomorphic or anti-isomorphic (by reversal of the operator).

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A027851 Number of nonisomorphic semigroups of order n.

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A058118 Triangle: self-converse semigroups of order n with k idempotents.

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A058106 Number of self-converse asymmetric 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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A186117 Number of nonisomorphic semigroups of order n minus number of groups of order n.

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