A076017 Number of nonisomorphic systems with n elements with one binary operation satisfying the equation B(AB)=A (semisymmetric quasigroups).

1, 1, 2, 3, 4, 9, 41, 595, 26620, 3908953, 1867918845

Offset: 1

Author

Richard C. Schroeppel, Oct 29 2002

Keywords

• nonn
• hard
• more

Comments

In January of 1968, Don Knuth described the concept of what he called an "abstract grope" to the students in his class for sophomore math majors at Caltech.

The students had just learned about abstract groups and he wanted them to get experience doing research with other algebraic axioms; so he challenged them to prove as many interesting things as they could about sets of elements with a binary operator that satisfies the identity x(yx)=y.

The name came from the fact that they were groping for results. Such systems were studied in a series of papers by Sade under a more complicated and more dignified yet less memorable name, "semisymmetric quasigroups". The students came up with some good stuff, including the concept of normal subgropes.

References

• D. E. Knuth, The Art of Computer Programming, Vol. 4B, in preparation.
• A. Sade, Quasigroupes demi-symétriques, Ann. Soc. Sci. Bruxelles Sér. I 79 (1965), 133-143.

Links

• Brendan D. McKay and Ian M. Wanless, Enumeration of Latin squares with conjugate symmetry, J. Combin. Des. 30 (2022), 105-130.

Crossrefs

Cf. A076016-A076021.

Extensions

a(10), a(11) and comments from Don Knuth, May 12 2005 - May 14 2005