A058108 -id:A058108 - cp's OEIS frontend

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

A058116 Triangle read by rows: T(n,k) is the number of isomorphism classes of commutative semigroups of order n with k idempotents.

Original entry on oeis.org

1, 2, 1, 5, 5, 2, 16, 23, 14, 5, 62, 106, 93, 49, 15, 342, 544, 582, 422, 200, 53, 3435, 3380, 3773, 3360, 2178, 943, 222, 97061, 30788, 27222, 26625, 21283, 12676, 5072, 1078

Offset: 1

Christian G. Bower, Nov 09 2000

			Triangle begins:
   1;
   2,   1;
   5,   5,  2;
  16,  23, 14,  5;
  62, 106, 93, 49, 15;
  ...

Index entries for sequences related to semigroups

Row sums give A001426.
Main diagonal is A006966(n+1).
Column 1 is A058117.
The labeled version is A058167.
Cf. A058108, A058142.

a(29)-a(36) from Andrew Howroyd, Jan 27 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

A058109 Semigroups of order n with 1 idempotent.

Original entry on oeis.org

1, 2, 5, 20, 171, 5284, 1224331

Offset: 1

Christian G. Bower, Nov 09 2000

Index entries for sequences related to semigroups

Column 1 of A058108.

Updated Feb 19 2001

A058111 Number of semigroups of order n with 3 idempotents.

Original entry on oeis.org

10, 72, 590, 5422, 61323

Offset: 3

Christian G. Bower, Nov 09 2000

Index entries for sequences related to semigroups

Column 3 of A058108.

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

A058110 Semigroups of order n with 2 idempotents.

Original entry on oeis.org

3, 9, 50, 309, 2806, 58583

Offset: 2

Christian G. Bower, Nov 09 2000

Index entries for sequences related to semigroups

Column 2 of A058108.

Updated Feb 19 2001

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

A027851 Number of nonisomorphic semigroups of order n.

Views

Author

Keywords

Links

Crossrefs

Formula

Extensions

A058116 Triangle read by rows: T(n,k) is the number of isomorphism classes of commutative semigroups of order n with k idempotents.

Views

Author

Keywords

Examples

Links

Crossrefs

Extensions

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

Views

Author

Keywords

Links

Crossrefs

Extensions

A058109 Semigroups of order n with 1 idempotent.

Views

Author

Keywords

Links

Crossrefs

Extensions

A058111 Number of semigroups of order n with 3 idempotents.

Views

Author

Keywords

Links

Crossrefs

Extensions

A118581 Number of nonisomorphic semigroups of order <= n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

Extensions

A118601 Partial sums of A058129.

Views

Author

Keywords

Crossrefs

Formula

Extensions

A058110 Semigroups of order n with 2 idempotents.

Views

Author

Keywords

Links

Crossrefs

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula