Constructing Double Magma with Commutation Operations

Abstract

A double magma is a nonempty set with two binary operations satisfying the interchange law. We call a double magma proper if the two operations are distinct and commutative if the operations are commutative. A double semigroup is a double magma for which both operations are associative. Given a group G we define a double magma (G,*,#) with the commutator operations x * y = [x,y] (= x^-1y^-1xy) 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. Note that the first law defines the variety of 3-metabelian groups. If both these laws hold in G, (G,*,#) is proper if and only if G contains a commutator whose square is nontrivial. B.H. Neumann has given an example of such a group which is not metabelian; thus the associated double magma is proper and produces an example with some complexity. The double magma (G,*,#) is a double semigroup if and only if G is nilpotent of class 2. In this case, (G,*,#) is a proper double semigroup if and only if G contains a commutator whose square is nontrivial. We construct a specific example letting G be the dihedral group of order 16. In addition we comment on a similar construction for rings using Lie commutators.