How to define a taxonomy of non associative operations? Let $A$ be a set, and let $a,b,c\in A$. Let also $\circ: A\times A\rightarrow A$ be a binary operation on $A$. We agree as usual to write $a\circ b$ to mean $\circ(a,b)$.
We say that $\circ$ is associative if for any $a,b,c\in A$ the following holds:
\begin{equation}
(a\circ b)\circ c=a\circ (b\circ c)
\end{equation}
That's OK, but given three elements in $A$ there are many ways to compose them. In particular, we can compute the following $12$ compositions:
\begin{equation}
(a\circ b)\circ c\qquad\qquad
a\circ (b\circ c)\\
(a\circ c)\circ b\qquad\qquad
a\circ (c\circ b)\\
(b\circ a)\circ c \qquad\qquad
b\circ (a\circ c) \\
(b\circ c) \circ a \qquad\qquad
b\circ (c \circ a) \\
(c\circ a)\circ b \qquad\qquad
c\circ (a\circ b) \\
(c\circ b)\circ a \qquad\qquad
c\circ (b\circ a)
\end{equation}
They are of course $12$ because we are permuting $3$ objects and for each permutation we have $2$ ways of putting meaningful parentheses, so $3!\cdot 2 = 12$. Associativity means that we identify the first (top left) object with the second (top right); this has a strong meaning to us because usual number fields ($\mathbb{R},\mathbb{C}$) behave in this way.
My question is: what happens if we identify the first relation with a different one taken from the other $11$? Are there some relevant (i.e. studied in literature) examples of such structures? The only examples I could find of non associative structures like this are defined over more complex objects, for example the cross product on $\mathbb{R}^3$ or generally a Lie bracket is not associative but is defined over a vector space. I don't even know what exactly to search for to organise my thoughts.
 A: If one drops the identity and associativity axioms from the definition of group, we get the definition of a quasigroup, and if we retain the identity axiom, the structure is called a loop. A priori these need not satisfy any further composition conditions, but some well-known nonassociative quasigroups do satisfy such conditions.
For example, the loop $\mathbb{O} - \{0\}$ of nonzero octonions (under multiplication) satisfies the Moufang identity:
$$(a \circ b) \circ (c \circ a) = a \circ ((b \circ c) \circ a),$$
and we call loops that satisfy this identity Moufang loops. All Moufang loops are alternative, which means that any subloop generated by two elements is associative, or equivalently, that $a \circ a \circ b, a \circ b \circ a, b \circ a \circ a$ are all unambiguous. (One can just as well ask for these identities to hold for a $k$-algebra.
As for Lie algebras (including the cross product on $\mathbb{R}^3$), they too satisfy a composition condition (which we can rearrange to via as a product rule for the map $a \circ \, \cdot \,$) called the Jacobi identity:
$$a \circ (b \circ c) + b \circ (c \circ a) + c \circ (a \circ b) = 0.$$
Probably without some additional hypotheses there is no sensible "taxonomy" of algebras satisfying another one of your other 10 quasi-associativity axiom (just for color, the number of general quasigroups of order $n$ grows very fast with $n$). Some of them lead to interesting conditions that can be described intuitively. For example, the condition $$(a \circ b) \circ c = c \circ (a \circ b)$$ just says that $\text{im } \circ$ is contained in the center of $(A, \circ)$; if $(A, \circ)$ is a quasigroup, then $\text{im }\circ = A$, so in that case the condition is equivalent to commutativity.
