Medial

Medial magmas

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. 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. Another class of semigroups forming medial magmas are the normal bands.[2] 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–Toyoda theorem

The Bruck–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.

Generalizations

The term medial or (more commonly) entropic is also used for a generalization to multiple operations. An algebraic structure is an entropic algebra[4] 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

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

References

1. Yamada, Miyuki (1971), "Note on exclusive semigroups", Semigroup Forum 3 (1): 160–167, doi:10.1007/BF02572956 .
2. Kuzʹmin, E. N. and Shestakov, I. P. (1995). "Non-associative structures". Algebra VI. Encyclopaedia of Mathematical Sciences 6. Berlin, New York: Springer-Verlag. pp. 197–280. ISBN 978-3-540-54699-3.
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