Constructing Double Magma with Commutation Operations

TL;DR

This paper explores the construction of double magmas from groups using commutator operations, characterizing when these structures are proper or semigroups, and provides explicit examples including dihedral groups.

Contribution

It introduces a novel construction of double magmas from groups via commutator operations and characterizes their properties in terms of group laws and examples.

Findings

Double magmas are constructed from groups using specific commutator operations.

The paper characterizes when these double magmas are proper or form double semigroups.

Explicit examples, including dihedral groups, illustrate the theoretical results.

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.