Moufang Loops

Abstract

We saw in Chapter I that there exist quasigroups (for example, the free quasigroups) possessing (multiplicative) homomorphisms upon systems which are not quasigroups — and similarity for loops. In order to avoid this possibility it is of interest (see 1.5, IV. 10) to consider a class, closed under homomorphism, of groupoids all of which are quasigroups. We may define such a class of groupoids G as follows. Let G be any groupoid with multiplication semigroup S such that for each x in G there exist θ, φ in S with R(x)θ = I, L(x) φ = I. Clearly every homo-morphic image of such a groupoid has the same property. Since S is generated by the set of all R (x), L (x) for x in G, every a in S has a β in S such that αβ = I; consequently, S is a group of permutations of G. Therefore G is a quasigroup.