View source for Distributive quasi-group

A [[Quasi-group|quasi-group]] which satisfies the left and the right distributive laws: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033580/d0335801.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033580/d0335802.png" />. In quasi-groups these two laws are independent of each other (there are left-distributive quasi-groups which are not right-distributive [[#References|[1]]]). As an example of a distributive quasi-group one may quote the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033580/d0335803.png" /> of rational numbers with the operation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033580/d0335804.png" />. Any idempotent medial quasi-group (i.e. a quasi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033580/d0335805.png" /> in which the relations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033580/d0335806.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033580/d0335807.png" /> are valid for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033580/d0335808.png" />) is distributive. In the general case every distributive quasi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033580/d0335809.png" /> is isotopic (cf. [[Isotopy|Isotopy]]) to a commutative [[Moufang loop|Moufang loop]] [[#References|[3]]]. Parastrophies (quasi-groups with respect to inverse operations, cf. [[Quasi-group|Quasi-group]]) of distributive quasi-groups are also distributive and are isotopic to the same commutative Moufang loop. If four elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033580/d03358010.png" /> in a distributive quasi-group are connected by the medial law: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033580/d03358011.png" />, 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 [[#References|[4]]].

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



====Comments====
In [[#References|[a1]]] it is shown that a quasi-group of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033580/d03358012.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033580/d03358013.png" /> distinct prime numbers and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033580/d03358014.png" /> non-negative integers, is isomorphic to the direct product of distributive quasi-groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033580/d03358015.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033580/d03358016.png" /> has order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033580/d03358017.png" /> and is Abelian (i.e. satisfies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033580/d03358018.png" />) for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033580/d03358019.png" />.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  V.M. Galkin,   "Finite distributive quasigroups"  ''Math. Notes'' , '''24'''  (1978)  pp. 525–526  ''Mat. Zametki'' , '''24'''  (1978)  pp. 39–41; 141</TD></TR></table>