Central Products of Cayley-Dickson Loops

Abstract

This paper studies the triviality of commutators in central products of Cayley-Dickson loops. Two immediate outcomes of this study are (1) the construction of a sequence of non-commutative di-associative loops in which the probability that a random commutator is trivial approaches 1, and (2) an easy proof that if two central products of $n$-fold Cayley-Dickson loops are isomorphic for $n\ge 3$, then the loops in the first product are term-wise isomorphic to the loops in the second product.