On decomposition of commutative Moufang groupoids
Novikov Boris · Quasigroups and Related Systems · 2008
We prove that every commutative Moufang groupoid is a semilattice of Archimedean subgroupoids. It is well-known that the multiplicative groupoid of an alternative/Jordan algebra satis es Moufang identities [1, 4]. Therefore it seems interesting to study the structure of such groupoids. In this note we apply to Moufang groupoids an approach which is widespread in semigroup theory decomposition into a semilattice of subsemigroups [3]. We shall call a groupoid with the identity (xy)(zx) = (x(yz))x (1) a Moufang groupoid. Everywhere in this article M denotes a commutative Moufang groupoid. Theorem 1. If M consists of idempotents, then it is a semilattice. Proof. Under assumption of the theorem it follows from (1) for y = z