A001329 -id:A001329 - cp's OEIS frontend

A118641 Number of nonisomorphic finite non-associative, invertible loops of order n.

1, 33, 2333

Offset: 5

Jonathan Vos Post, May 10 2006

These are non-associative loops in which every element has a unique inverse and it includes IP, Moufang and Bol loops [Cawagas]. The data were generated and checked by a supercomputer of 48 Pentium II 400 processors, specially built for automated reasoning, in about three days. In general, a loop is a quasigroup with an identity element e such that xe = x and ex = x for any x in the quasigroup. All groups are loops. A quasigroup is a groupoid G such that for all a and b in G, there exist unique c and d in G such that ac = b and da = b. Hence a quasigroup is not required to have an identity element, nor be associative. Equivalently, one can state that quasigroups are precisely groupoids whose multiplication tables are Latin squares (possibly empty).

			a(5) = 1 (which is non-Abelian).
a(6) = 33 (7 Abelian + 26 non-Abelian).
a(7) = 2333 (16 Abelian + 2317 non-Abelian).

Cawagas, R. E., Terminal Report: Development of the Theory of Finite Pseudogroups (1998). Research supported by the National Research Council of the Philippines (1996-1998) under Project B-88 and B-95.

Hantao Zhang, Generation of NAFILs of Order 7, Association for Automated Reasoning, No. 46, 2000.
John Pedersen, Loops.

Cf. A001329.

A350876 a(n) is the number of nonisomorphic flexible left-right-alternative magmas with n elements. That is, a(n) is the number of nonisomorphic magmas with n elements which satisfy all of the identities x(yx) = (xy)x, (xx)y = x(xy), and x(yy) = (xy)y (for all x and y).

Original entry on oeis.org

1, 1, 5, 33, 675, 65066, 41160471

Offset: 0

Joel Brennan, Jan 23 2022

Compare A350873 and A350875, which are the numbers of flexible magmas with n elements and left-right-alternative magmas with n elements (up to isomorphism). The fact that a(n) < A350875(n) for n >= 3 means that left-right-alternativity for magmas (the identities (xx)y = x(xy) and x(yy) = (xy)y) does not imply flexibility (x(yx) = (xy)x). This is in contrast to the situation for non-associative rings, where this implication does hold (due to the additional additive structure).

			There are 10 nonisomorphic magmas with 2 elements, 5 of which are flexible and left-right-alternative, so a(2) = 5.
Similarly there are 3330 nonisomorphic magmas with 3 elements, 33 of which satisfy all of (xy)x = x(yx), (xx)y = x(xy), and x(yy) = (xy)y for all x and y, so a(3) = 33.

Cf. A001329 (magmas), A350873 (flexible magmas), A350874 (left/right-alternative magmas), A350875 (left-right-alternative magmas).

a(5)-a(6) from Andrew Howroyd, Jan 25 2022

A362384 Number of nonisomorphic magmas with n elements satisfying the equation x(yz) = xz.

Original entry on oeis.org

1, 1, 4, 12, 81, 934, 23703, 1219177

Offset: 0

Andrew Howroyd, Apr 24 2023

Cf. A001329 (magmas), A279644 (labeled case), A362385.

A030246 Number of nonisomorphic groupoids with a nontrivial symmetry.

Original entry on oeis.org

0, 0, 4, 93, 49627

Offset: 0

Christian G. Bower

Index entries for sequences related to groupoids

A001329 - A030245.

A038016 Number of pointed (distinguished element) commutative groupoids.

Original entry on oeis.org

1, 8, 369, 175296, 1271911125, 182810647581408, 638870563764553089382, 64388603066507462072885127552, 216467350495255331602790475962162008809, 27557319224243445068662283861406911750848082900080

Offset: 1

Christian G. Bower, May 15 1998

nonn

Index entries for sequences related to groupoids

Cf. A001329, A001425.

A038019 Triangle: T(n,k), k<=n: groupoids with no symmetry with n elements and k idempotents.

Original entry on oeis.org

1, 0, 1, 1, 4, 1, 969, 1431, 729, 108, 56615456, 75491208, 37741600, 8386560, 697501

Offset: 0

Christian G. Bower, May 15 1998

Index entries for sequences related to groupoids

Cf. A001329, A030248, A030251, A030271.

A038020 Triangle: T(n,k), k<=n: groupoids with a nontrivial symmetry with n elements and k idempotents.

Original entry on oeis.org

0, 0, 0, 2, 0, 2, 9, 54, 0, 30, 15376, 12664, 14304, 4096, 3187

Offset: 0

Christian G. Bower, May 15 1998

			Triangle begins:
    0;
    0,     0;
    2,     0,     2;
    9,    54,     0,   30;
15376, 12664, 14304, 4096, 3187;
  ...

Index entries for sequences related to groupoids

Row sums are A030246.
Main diagonal is A030249.
Columns k=0..1 are A030252, A030254.
Cf. A001329, A030248, A030251, A030271.

Difference of A038018 and A038019.

A038022 Triangle: T(n,k), k<=n: commutative groupoids with no symmetry with n elements and k idempotents.

Original entry on oeis.org

1, 0, 1, 1, 2, 1, 35, 51, 27, 3, 13776, 18330, 9152, 2016, 164, 83306272, 104124850, 52056277, 13006746, 1622527, 80461

Offset: 0

Christian G. Bower, May 15 1998

Index entries for sequences related to groupoids

Cf. A001329, A001425, A030258, A030261, A030264.

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

A118641 Number of nonisomorphic finite non-associative, invertible loops of order n.

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A350876 a(n) is the number of nonisomorphic flexible left-right-alternative magmas with n elements. That is, a(n) is the number of nonisomorphic magmas with n elements which satisfy all of the identities x(yx) = (xy)x, (xx)y = x(xy), and x(yy) = (xy)y (for all x and y).

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A362384 Number of nonisomorphic magmas with n elements satisfying the equation x(yz) = xz.

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A030246 Number of nonisomorphic groupoids with a nontrivial symmetry.

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A038016 Number of pointed (distinguished element) commutative groupoids.

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A038019 Triangle: T(n,k), k<=n: groupoids with no symmetry with n elements and k idempotents.

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A038020 Triangle: T(n,k), k<=n: groupoids with a nontrivial symmetry with n elements and k idempotents.

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A038022 Triangle: T(n,k), k<=n: commutative groupoids with no symmetry with n elements and k idempotents.

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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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