# Distributive quasi-group

A quasi-group which satisfies the left and the right distributive laws: $x\cdot yz = xy \cdot xz$, $yz \cdot x = yx \cdot zx$. In quasi-groups these two laws are independent of each other (there are left-distributive quasi-groups which are not right-distributive ). As an example of a distributive quasi-group one may quote the set $\mathbf{Q}$ of rational numbers with the operation $x \cdot y = (x+y)/2$. Any idempotent medial quasi-group (i.e. a quasi-group $Q$ in which the relations $x\cdot x = x$ and $xy \cdot uv = xu \cdot yv$ are valid for all $x,y,u,v$) is distributive. In the general case every distributive quasi-group $(Q,{\cdot})$ is isotopic to a commutative Moufang loop . Parastrophies (quasi-groups with respect to inverse operations, cf. Quasi-group) of distributive quasi-groups are also distributive and are isotopic to the same commutative Moufang loop. If four elements $a,b,c,d$ in a distributive quasi-group are connected by the medial law: $ab\cdot cd = ac \cdot bd$, they generate a medial sub-quasi-group. In particular, any three elements of a distributive quasi-group generate a medial sub-quasi-group. In a sub-quasi-group the translations are automorphisms, and in a certain sense a distributive quasi-group is homogeneous: no element, and no sub-quasi-group, is distinguished. The group generated by all right translations of a finite distributive quasi-group is solvable .