Distributive quasi-group

A quasi-group which satisfies the left and the right distributive laws: , . 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 of rational numbers with the operation . Any idempotent medial quasi-group (i.e. a quasi-group in which the relations and are valid for all ) is distributive. In the general case every distributive quasi-group is isotopic (cf. Isotopy) 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 in a distributive quasi-group are connected by the medial law: , 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 , with distinct prime numbers and non-negative integers, is isomorphic to the direct product of distributive quasi-groups , where has order and is Abelian (i.e. satisfies ) for .

References

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