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

1, 33, 2333

Offset

5,2

Comments

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).

References

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.

Links

Table of n, a(n) for n=5..7.

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

John Pedersen, Loops.

Example

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

Crossrefs

Cf. A001329.

Sequence in context: A294773 A294611 A294954 * A263908 A358808 A111922

Adjacent sequences: A118638 A118639 A118640 * A118642 A118643 A118644

Keyword

nonn,bref,more

Author

Jonathan Vos Post, May 10 2006

Status

approved