INTERCHANGE RINGS

An interchange ring,$(R,+,\bullet )$, is an abelian group with a second binary operation defined so that the interchange law$(w+x)\bullet (y+z)=(w\bullet y)+(x\bullet z)$ holds. An interchange near ring is the same structure based on a group which may not be abelian. It is shown that each interchange (near) ring based on a group $G$ is formed from a pair of endomorphisms of $G$ whose images commute, and that all interchange (near) rings based on $G$ can be characterized in this manner. To obtain an associative interchange ring, the endomorphisms must be commuting idempotents in the endomorphism semigroup of $G$. For $G$ a finite abelian group, we develop a group-theoretic analogue of the simultaneous diagonalization of idempotent linear operators and show that pairs of endomorphisms which yield associative interchange rings can be diagonalized and then put into a canonical form. A best possible upper bound of $4^{r}$ can be given for the number of distinct isomorphism classes of associative interchange rings based on a finite abelian group $A$ which is a direct sum of $r$ cyclic groups of prime power order. If $A$ is a direct sum of $r$ copies of the same cyclic group of prime power order, we show that there are exactly ${\textstyle \frac{1}{6}}(r+1)(r+2)(r+3)$ distinct isomorphism classes of associative interchange rings based on $A$. Several examples are given and further comments are made about the general theory of interchange rings.

Constructing Double Magma on Groups Using Commutation Operations

A 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.