-groups, diagonalizable automorphisms and loops
Some results are obtained for quasigroups in which the operation of iterated squaring always leads to stability irrespective of the initial element.
We consider the question of which loops are capable of expressing arbitrary Boolean functions through expressions of constants and variables. We call this property Boolean completeness. It is a generalization of functional completeness, and is intimately connected to the computational complexity of various questions about expressions, circuits, and equations defined over the loop. We say that a loop is polyabelian if it is an iterated affine quasidirect product of Abelian groups; polyabelianness...
The paper surveys the known results concerning the question: “For what values of does there exist a nonassociative Moufang loop of order ?” Proofs of the newest results for odd, and a complete resolution of the case even are also presented.
A groupoid is alternative if it satisfies the alternative laws and . These laws induce four partial maps on