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

1, 33, 2333

Offset: 5

Author

Jonathan Vos Post, May 10 2006

Keywords

• nonn
• bref
• more

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).

Examples

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

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

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

Crossrefs

Cf. A001329.