Medial magma

In abstract algebra, a medial magma or medial groupoid is a magma or groupoid (that is, a set with a binary operation) that satisfies the identity

(x • y) • (u • v) = (x • u) • (y • v),

or more simply,

xy • uv = xu • yv

for all x, y, u and v, using the convention that juxtaposition denotes the same operation but has higher precedence. This identity has been variously called medial, abelian, alternation, transposition, interchange, bi-commutative, bisymmetric, surcommutative, entropic, etc.^[1]

Any commutative semigroup is a medial magma, and a medial magma has an identity element if and only if it is a commutative monoid. The "only if" direction is the Eckmann–Hilton argument. Another class of semigroups forming medial magmas are normal bands.^[2] Medial magmas need not be associative: for any nontrivial abelian group with operation + and integers m ≠ n, the new binary operation defined by x • y = mx + ny yields a medial magma that in general is neither associative nor commutative.

Using the categorical definition of product, for a magma M, one may define the Cartesian square magma M × M with the operation

(x, y) • (u, v) = (x • u, y • v).

The binary operation • of M, considered as a mapping from M × M to M, maps (x, y) to x • y, (u, v) to u • v, and (x • u, y • v)  to (x • u) • (y • v) . Hence, a magma M is medial if and only if its binary operation is a magma homomorphism from M × M to M. This can easily be expressed in terms of a commutative diagram, and thus leads to the notion of a medial magma object in a category with a Cartesian product. (See the discussion in auto magma object.)

If f and g are endomorphisms of a medial magma, then the mapping f • g defined by pointwise multiplication

(f • g)(x) = f(x) • g(x)

is itself an endomorphism. It follows that the set End(M) of all endomorphisms of a medial magma M is itself a medial magma.

Bruck–Murdoch–Toyoda theorem

The Bruck–Murdoch–Toyoda theorem provides the following characterization of medial quasigroups. Given an abelian group A and two commuting automorphisms φ and ψ of A, define an operation • on A by

x • y = φ(x) + ψ(y) + c,

where c some fixed element of A. It is not hard to prove that A forms a medial quasigroup under this operation. The Bruck–Toyoda theorem states that every medial quasigroup is of this form, i.e. is isomorphic to a quasigroup defined from an abelian group in this way.^[3] In particular, every medial quasigroup is isotopic to an abelian group.

The result was obtained independently in 1941 by Murdoch and Toyoda.^[4][5] It was then rediscovered by Bruck in 1944.^[6]

Generalizations

The term medial or (more commonly) entropic is also used for a generalization to multiple operations. An algebraic structure is an entropic algebra^[7] if every two operations satisfy a generalization of the medial identity. Let f and g be operations of arity m and n, respectively. Then f and g are required to satisfy

${\displaystyle f(g(x_{11},\ldots ,x_{1n}),\ldots ,g(x_{m1},\ldots ,x_{mn}))=g(f(x_{11},\ldots ,x_{m1}),\ldots ,f(x_{1n},\ldots ,x_{mn})).}$

Nonassociative examples

A particularly natural example of a nonassociative medial magma is given by collinear points on elliptic curves. The operation x • y = −(x + y) for points on the curve, corresponding to drawing a line between x and y and defining x • y as the third intersection point of the line with the elliptic curve, is a (commutative) medial magma which is isotopic to the operation of elliptic curve addition.

Unlike elliptic curve addition, x • y is independent of the choice of a neutral element on the curve, and further satisfies the identities x • (x • y) = y. This property is commonly used in purely geometric proofs that elliptic curve addition is associative.

Citations

References

This page was last edited on 26 March 2024, at 13:08