Some questions on interdependence of some properties of abstract magma 
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*Does there exist a magma $(S,\cdot)$ such that for every $y\in S, \exists y'\in S$ such  that $x\cdot(y\cdot y')=x, \forall x,y\in S$, but there exist $x_1, x_2, x_3\in S$ such that $x_1\cdot(x_2\cdot x_3)\neq(x_2\cdot x_1)\cdot x_3$?

*Does there exist a magma $(S,\cdot)$ such that for every $y\in S, \exists y'\in S$ such  that $x\cdot(y'\cdot y)=x, \forall x,y\in S$, but there exist $x_1, x_2, x_3\in S$ such that $x_1\cdot(x_2\cdot x_3)\neq(x_2\cdot x_1)\cdot x_3$?

*Does there exist a magma $(S,\cdot)$ such that for every $a,b\in S$, $\exists x\in S$ such that $x\cdot a=b$, but there exist $x_1, x_2, x_3\in S$ such that $x_1\cdot(x_2\cdot x_3)\neq(x_2\cdot x_1)\cdot x_3$?
 A: The groupoids you described in 1. and 2. look somehow similar to (left/right) inverse property groupoids, but they are not the same. I think is possible to find examples of non-associative magmas veryfying exactly your identities. I'll search and post if I find some.

We can say a bit more about 3.
A magma $(S,\cdot)$ such that $\forall\,a,b\in S$, $\exists!\,x\in S$ such that $x\cdot a=b$ is called right quasigroup.
A magma $(S,\cdot)$ such that $\forall\,a,b\in S$, $\exists!\,x\in S$ such that $a\cdot x=b$ is called left quasigroup.
If you change $\exists!$ to $\exists$ in the latter two, then right (left) translation on $S$ are not bijective but are only onto $S$. You call such structures right/left division groupoid.
In this article, On definitions of groupoids closely connected with quasigroups by V.A. Shcherbacov, you find definitions of translations and definitions in such terms of some magmas you may be interested in.

Addendum:
Check pages 5-6 of this other paper, The varieties of quasigroups of bol-moufang type:
an equational reasoning approach (Phillips-Vojtechovsky). This is not yet what you ask but gives you an overview of identities in magmas. I want to underline that those listed in this latter are properties weaker than associative, while - to do best - you should search instead properties related to left/right inverse property in non-associative groupoids. It is also important to specify "in non-associative groupoids" because, in groups, many identities similar to yours are trivial.
Post updates if you have any, I'll do the same!
