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 [1]). 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 [3]. 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 [4].

References

[1]	Sh. Stein, "On a construction of Hosszú" Publ. Math. Debrecen , 6 : 1–2 (1959) pp. 10–14
[2]	V.D. Belousov, "The structure of distributive quasigroups" Mat. Sb. , 50 : 3 (1960) pp. 267–298 (In Russian)
[3]	V.D. Belousov, "Foundations of the theory of quasi-groups and loops" , Moscow (1967) (In Russian)
[4]	B. Fischer, "Distributive Quasigruppen endlicher Ordnung" Math. Z. , 83 : 4 (1964) pp. 267–303

Comments

In [a1] it is shown that a quasi-group of order $p_1^{\alpha_1}\cdots p_k^{\alpha_k}$, with $p_1,\ldots,p_k$ distinct prime numbers and $\alpha_1,\ldots,\alpha_k$ non-negative integers, is isomorphic to the direct product of distributive quasi-groups $Q_1,\ldots,Q_k$, where $Q_i$ has order $p_i^{\alpha_i}$ and is medial (i.e. satisfies $ab\cdot cd = ac \cdot bd$) for $p_1 \ne 3$.

References

[a1]	V.M. Galkin, "Finite distributive quasigroups" Math. Notes , 24 (1978) pp. 525–526 Mat. Zametki , 24 (1978) pp. 39–41; 141