Constructing Double Magma on Groups Using Commutation Operations

Abstract

AbstractA magma (M, *) is a nonempty set with a binary operation. A double magma (M, *, •) is a nonempty set with two binary operations satisfying the interchange law (w * x) • (y * z) = (w • y)*(x•z). We call a double magma proper if the two operations are distinct, and commutative if the operations are commutative. A double semigroup, first introduced by Kock, is a double magma for which both operations are associative. Given a non-trivial group G we define a system of two magma (G, *, •) using the commutator operations x * y = [x, y](= x−1 y−1x y) and x • y = [y, x]. We show that (G, *, •) is a double magma if and only if G satisfies the commutator laws [x, y; x, z] = 1 and [w, x; y, z]2 = 1. We note that the first lawdefines the class of 3-metabelian groups. If both these laws hold in G, the double magma is proper if and only if there exist x0, y0 ∊ G for which [x0 , y0]2 ≠ 1. This double magma is a double semigroup if and only if G is nilpotent of class two. We construct a specific example of a proper double semigroup based on the dihedral group of order 16. In addition, we comment on a similar construction for rings using Lie commutators.