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I'm wondering what kind of structures don't contain associativity in the set of its axioms. Are they useful? Where?

For example, $\left(\mathbb{Z},-\right)$ and $\left(\mathbb{Q}\setminus\{0\},:\right)$ are pretty good structures which are closed, have identity elements and even inverses (here : is division). Although I don't like them, because they are just "generated" from well-known groups, and I've just realized that we can similarly create such structures from any group sending $a\star b$ to $ab^{-1}$. Funny, but I can't find other examples which could be interesting, except the non-natural ones.

One of the nice things about this kind of structures, for example, a set with closed operation, identity and unique inverse for every element, is that it's possible to calculate the number of non-isomorphic structures of finite fixed order. Another thing is that we can easily check the axioms from Cayley table. That's all, I suppose.

Mihail
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    Try looking up Lie algebras. – Max Sep 10 '15 at 20:10
  • The octonions and split-octonions are both important examples in their own right, and are the source of the many more. Beyond their nonassociativity, they are deeply interesting in part because of their connection with other exceptional objects, e.g., the exceptional simple Lie algebras. – Travis Willse Sep 10 '15 at 21:24
  • An example that already familiar is the standard cross product $\times$ on $\Bbb R^3$. It is linear and antisymmetric, but instead of associativity it satisfies the Jacobi identity ${\bf x} \times ({\bf y} \times {\bf z}) + {\bf y} \times ({\bf z} \times {\bf x}) + {\bf z} \times ({\bf x} \times {\bf y}) = 0$, which makes it a Lie algebra, and thus a special case of a general example mentioned here; in fact, it is isomorphic to the simple Lie algebra $\frak{so}(3)$. – Travis Willse Sep 10 '15 at 21:26
  • Also look up quasigroups! Have a look at my answer and please comment if you're interested in going deeper into it :) – MattAllegro Apr 09 '20 at 20:15

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There are many kinds of non-associative algebras, arising from geometry, physics and other areas. Examples are Lie algebras, Jordan algebras, Leibniz algebras, and left-symmetric algebras. The latter algebras appear in the classification of convex homogeneous cones, in operad theory (as the operad of rooted trees), in the theory of affinely flat manifolds, in vertex algebras, in renormalisation theory and many other areas - see the survey here, which hopefully shows that they are "useful".

Dietrich Burde
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Indeed $(\mathbb{Z},-)$ and $(\mathbb{Q}\backslash\{0\},:)$ are quasigroups obtained from groups: this article is in Italian but equation (1) in it doesn't lie: $$a\circ b=ab^{-1}.$$ And quasigroups are groupoids (or magmas) with two additional properties, called cancellation and division.

MattAllegro
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