Medial magma

For the triple product, see Median algebra.

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

$(x \cdot y) \cdot (u \cdot v) = (x \cdot u) \cdot (y \cdot v)$, or more simply, $xy\cdot uv = xu\cdot yv$

using the convention that juxtaposition denotes the same operation but has higher precedence. A magma or groupoid is an algebraic structure that generalizes a group. This identity has been variously called medial, abelian, alternation, transposition, interchange, bi-commutative, bisymmetric, surcommutative, entropic etc.

Any commutative semigroup is a medial magma, and a medial magma has an identity element if and only if it is a commutative monoid. Another class of semigroups forming medial magmas are the normal bands. Medial magmas need not be associative: for any nontrivial abelian group and integers m ≠ n, replacing the group operation $x+y$ with the binary operation $x \cdot y = mx+ny$ yields a medial magma which in general is neither associative nor commutative.

Using the categorial definition of the product, 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 function on M × 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

$(f\cdot g)(x) = f(x)\cdot g(x)$

is itself an endomorphism.

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. In particular, every medial quasigroup is isotopic to an abelian group.

The result was obtained independently in 1941 by D.C. Murdoch and K. Toyoda. It was then rediscovered by Bruck in 1944.

Generalizations

The term medial or (more commonly) entropic is also used for a generalization to multiple operations. An algebraic structure is an entropic algebra 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

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

See also

• Category of medial magmas

References

• D.C. Murdoch, Structure of abelian quasigroups. Trans. Amer. Math. Soc, 1941, 47, p. 134-138.
• K. Toyoda, On axioms of linear functions. Proc. Imp. Acad. Tokyo, 1941, 17, p. 221-227
• R. H. Bruck, Some results in the theory of quasigroups, Trans. Amer. Math. Soc, 1944, 55.