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%I #16 Nov 28 2023 08:58:38
%S 1,33,2333
%N Number of nonisomorphic finite non-associative, invertible loops of order n.
%C 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).
%D 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.
%H Hantao Zhang, <a href="http://aarinc.org/Newsletters/046-2000-01.html#nafil">Generation of NAFILs of Order 7</a>, Association for Automated Reasoning, No. 46, 2000.
%H John Pedersen, <a href="http://www.math.usf.edu/~eclark/algctlg/loops.html">Loops.</a>
%e a(5) = 1 (which is non-Abelian).
%e a(6) = 33 (7 Abelian + 26 non-Abelian).
%e a(7) = 2333 (16 Abelian + 2317 non-Abelian).
%Y Cf. A001329.
%K nonn,bref,more
%O 5,2
%A _Jonathan Vos Post_, May 10 2006
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