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I am looking for examples of (small) finite loops and monoids that are not groups for demonstrating what happens if you omit some of the group axioms.

Does anyone know some ressources for this? I already used Wikipedia and Google but the problem I encountered was that I could not find any examples that were not groups.

MattAllegro
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flawr
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    Hi! Could you be more specific than what happens if? Which properties are you mainly interested in? I will update my answer with more resources consequently... – MattAllegro Apr 19 '15 at 10:44
  • @MattAllegro Thank you for your answer so far! I am not looking for any special property, I am just looking for some examples, that are not groups but very 'close' to being groups. I'd just like to find 'small' examples, so the multiplication table does not get too big=) – flawr Apr 19 '15 at 10:50
  • Ok, seen. Keep an eye over this page in the next days, I'll update adding monoids also. Another resource I've cited here multiple times is: http://www.quasigroups.eu/contents/download/2011/19_1.pdf – MattAllegro Apr 19 '15 at 10:54
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    Thank you very much! I really appreciate your help! – flawr Apr 19 '15 at 10:55
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    Monoids: have you checked the multiplicative structures over $\Bbb Z_n$ or over the matrices of finite given order with entries in a finite field? – MattAllegro Apr 19 '15 at 14:04
  • @MattAllegro No, thank you for the hint! You mean equipping $\mathbb Z_n$ with the multiplication $\mod n$ where $n \neq p^m$? – flawr Apr 19 '15 at 14:50
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    Yes, exactly. Start with this example if you're interested in the multiplication table. – MattAllegro Apr 19 '15 at 14:52
  • @MattAllegro: I just tried it successfully: Write down the table of $R\setminus {0}$ and delete all rows/columns that contain a $0$. If there still are rows/columns with twice the same entry, we are finished! Works e.g. for $n=10$, where you end up removing the $5$-row/column. – flawr Apr 19 '15 at 20:12
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    Travis Willse asks a related MSE question. – Peter Kagey May 10 '22 at 23:35
  • @PeterKagey Thanks for the link! – flawr May 11 '22 at 11:22

2 Answers2

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Several examples of finite loops (quasigroups with left and right identity) of small order with some additional properties can be found here.

Another source showing several examples of finite loops Cayley tables.

A third example of very group-like loop which is not a group, answer to a related Math.SE question.

Again, in the absence of associativity, inverses may be defined in a more general way: this is the case of inverse property loops. The smallest example of such loop which is not a group has order $7$. This latter is, to me, a brilliant example of what may happen if we omit some of the group axioms (associativity, in the case of loops).

MattAllegro
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Another example. If $\mathbb O$ is an algebra of octonions with a standard basis $e_0,\ldots,e_7$, then $L=\{\pm e_0,\pm e_1,\ldots,\pm e_7\}$ is a finite non-associative and non-abelian loop with elegant law of composition (Fano plane).

Alex W
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