Non-associative algebraic structures 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.

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".
