Exercise 6, § 3

An internal law of composition $\top$ on $E$ is said to determine on $E$ a quasi-group structure if, for all $x \in E$, the left and right translations $\gamma_x$ and $\delta_x$ are bijective mappings of $E$ onto itself. A quasi-group is said to be distributive if the law $\top$ is doubly distributive with respect to itself.

(a) Determine all the distributive quasi-group structures on a set of $n$ elements for $2 \leq n \leq 6$.

(b) Show that the set $\mathbf{Q}$ of rational numbers, with the law of composition $(x, y) \mapsto \frac{1}{2}(x + y)$, is a distributive quasi-group.

(c) In a distributive quasi-group $E$, every element is idempotent. Deduce that, if E has more than one element, the law $\top$ can neither have an identity element nor be associative.

(d) The right and left translations of a distributive quasi-group E are automorphisms of E.

(e) If E is a finite distributive quasi-group, the structure induced on every stable subset of E is a distributive quasi-group structure.

(f) Let E be a distributive quasi-group. If R is an equivalence relation which is left (resp. right) compatible (no. 3) with the law $\top$, the classes mod. R are stable subsets of E. If E is finite, all these classes are obtained one from another by left (resp. right) translation; under the same conditions, if R is compatible with the law $\top$, the quotient structure on E/R is a distributive quasi-group structure.

(g) Let E be a distributive quasi-group. The set $A_a$ of elements of E which are permutable with a given element a is stable; if E is finite, for all $x \in A_a$, $A_x = A_a$ (note, using (e), that there exists $y \in A_a$ such that $x = a \top y$) and when x runs through E the sets $A_x$ are identical with the equivalence classes with respect to a relation compatible with the law $\top$.

(h) If E is a finite distributive quasi-group and the law $\top$ is commutative, the number of elements of E is odd (consider the ordered pairs $(x, y)$ of elements of E such that $x \top y = y \top x = a$, where a is given).