5,2

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).

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.

Pedersen, J., Loops.

a(5) = 1 (which is nonabelian).

a(6) = 33 (7 Abelian + 26 nonabelian).

a(7) = 2333 (16 Abelian + 2317 nonabelian).

Cf. A001329.

Sequence in context: A093756 A120288 A099370 this_sequence A111922 A136541 A114071

Adjacent sequences: A118638 A118639 A118640 this_sequence A118642 A118643 A118644

bref,nonn

Jonathan Vos Post (jvospost3(AT)gmail.com), May 10 2006