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How many finite loops of order $n$ are there?

I am interested in the exact values ​​of $n$ if $n <40$ or even reasonable estimates. I am also interested in formulae or bounds for all $n$.

Note that similar questions have been studied for groups and semigroups. Much information is available in the group theory case, but even for semigroups not much is known beyond $n=11$.

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    $\begingroup$ You need to make the question self-contained (so that the title is not needed to make sense of the question). And please introduce all notation used. $\endgroup$
    – Tobias Kildetoft
    Commented Dec 16, 2013 at 11:39
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    $\begingroup$ Loop - en.wikipedia.org/wiki/Loop_%28algebra%29 ; Formulae, if it does not, then the asymptotics. $\endgroup$
    – user44255
    Commented Dec 16, 2013 at 11:58
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    $\begingroup$ Trivial? Number quasigroups of order greater than 11 do not know how I know. $\endgroup$
    – user44255
    Commented Dec 16, 2013 at 12:17
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    $\begingroup$ @TobiasKildetoft, I seriously doubt you can find these numbers on a computer. For many years they couldn't compute the number of semigroups of order 9 on a computer. Now I think due to some conceptual advances on counting nilpotents they may be as far as 11. It wouldn't surprise me if the number of loops of order 40 is much higher than the US debt or even the number of atoms in the universe. $\endgroup$
    – Benjamin Steinberg
    Commented Dec 16, 2013 at 12:18
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    $\begingroup$ I think this question is not totally unreasonable. People write nontrivial papers to compute the number of semigroups of order 10. $\endgroup$
    – Benjamin Steinberg
    Commented Dec 16, 2013 at 12:19

2 Answers 2

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As @Benjamin said in comments above, this is a very difficult question. However if one is willing to add conditions, then there are some relevant results. Firstly:

Chein, Orin, * Moufang Loops of Small Order*, Trans. Amer. Math. Soc. 188 (1974), 31-51.

Chein completely classifies all non-associative Moufang loops of order $\leq 31$. There are 13 such loops - one of order 12, five of order 16, one of order 20, five of order 24, and one of order 28.

In fact Chein extends this classification to give a full classification up to order $\leq 63$ in:

Chein, Orin, Moufang loops of small order, Mem. Amer. Math. Soc. 13 (1978), no. 197, iv+131 pp.

You may also be interested in this webpage which gives a full list of all Bol loops of order $\leq 31$.

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Here you can find the number of loops of order n up to isotopy and isomorphism for n at most 10, and the number of quasigroups up to isomorphism.

http://cs.anu.edu.au/~bdm/papers/ls_final.pdf

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