A350875 -id:A350875 - cp's OEIS frontend

A350874 a(n) is the number of nonisomorphic magmas with n elements which satisfy the identity (xx)y = x(xy) for all x and y (so-called left-alternative magmas).

1, 1, 5, 97, 25311, 165974649

Offset: 0

Joel Brennan, Jan 20 2022

Equivalently (by symmetry), a(n) also equals the number of nonisomorphic right-alternative magmas with n elements (that is, magmas satisfying the identity x(yy) = (xy)y for all x and y).

			There are 10 non-isomorphic magmas with 2 elements, and 5 of these are left-alternative, so a(2) = 5.

Cf. A001329 (magmas), A350875 (left-right-alternative magmas), A350876, A350873.

a(5) from Andrew Howroyd, Jan 29 2022

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

A384190 Number of non-isomorphic AG-groupoids of order n.

Original entry on oeis.org

1, 3, 20, 331, 31913, 40104513, 643460323187

Offset: 1

Elijah Beregovsky, May 21 2025

A magma S is called an Abel-Grassmann or AG-groupoid (historically they were also called left almost semigroups, right modular groupoids and left invertive groupoids) if for all a,b,c in S (ab)c = (cb)a.

			For a(2) there are only 3 non-isomorphic AG-groupoids: the null semigroup, the semigroup formed by the set {0,1} under multiplication and the cyclic group Z2.

M. A. Kazim and M. Naseerudin, On almost semigroups, Alig. Bull. Math. 2, 1-7 (1972).

Marek Dančo, Mikoláš Janota, Michael Codish, and João Jorge Araújo, Complete Symmetry Breaking for Finite Models, arXiv:2502.10155 [cs.LO], 2025.
Marek Dančo, MACE-like model finder for algebraic structures.
Andreas Distler, Muhammad Shah, and Volker Sorge, Enumeration of AG-Groupoids, Davenport, J.H., Farmer, W.M., Urban, J., Rabe, F. (eds) Intelligent Computer Mathematics. CICM 2011. Lecture Notes in Computer Science, vol 6824. Springer, Berlin, Heidelberg.
Muhammad Iqbal, Imtiaz Ahmad, Muhammad Shah, and Muhammad Irfan Ali, On cyclic associative Abel-Grassman groupoids, arXiv:1510.01316 [math.GR], 2015.
Index entries for sequences related to groupoids

Cf. A001329 (magmas), A124506 (semigroups), A001426, A350875, A350874.

cp's OEIS Frontend

A350874 a(n) is the number of nonisomorphic magmas with n elements which satisfy the identity (xx)y = x(xy) for all x and y (so-called left-alternative magmas).

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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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A384190 Number of non-isomorphic AG-groupoids of order n.

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