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
or more simply,
for all Template:Math, Template:Math, Template:Math and Template:Math, 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.Template:Sfn
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.Template:Sfn Medial magmas need not be associative: for any nontrivial abelian group with operation Template:Math and integers Template:Math, the new binary operation defined by Template:Math yields a medial magma that in general is neither associative nor commutative.
Using the categorical definition of product, for a magma Template:Math, one may define the Cartesian square magma Template:Math with the operation
The binary operation Template:Math of Template:Math, considered as a mapping from Template:Math to Template:Math, maps Template:Math to Template:Math, Template:Math to Template:Math, and Template:Math to Template:Math. Hence, a magma Template:Math is medial if and only if its binary operation is a magma homomorphism from Template:Math to Template:Math. 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 Template:Math and Template:Math are endomorphisms of a medial magma, then the mapping Template:Math defined by pointwise multiplication
is itself an endomorphism. It follows that the set Template:Math of all endomorphisms of a medial magma Template:Math is itself a medial magma.
Bruck–Murdoch–Toyoda theoremEdit
The Bruck–Murdoch–Toyoda theorem provides the following characterization of medial quasigroups. Given an abelian group Template:Math and two commuting automorphisms Template:Math and Template:Math of Template:Math, define an operation Template:Math on Template:Math by
where Template:Math some fixed element of Template:Math. It is not hard to prove that Template:Math forms a medial quasigroup under this operation. The Bruck–Murdoch-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.Template:Sfn In particular, every medial quasigroup is isotopic to an abelian group.
The result was obtained independently in 1941 by Murdoch and Toyoda.Template:SfnTemplate:Sfn It was then rediscovered by Bruck in 1944.Template:Sfn
GeneralizationsEdit
The term medial or (more commonly) entropic is also used for a generalization to multiple operations. An algebraic structure is an entropic algebraTemplate:Sfn if every two operations satisfy a generalization of the medial identity. Let Template:Math and Template:Math be operations of arity Template:Math and Template:Math, respectively. Then Template:Math and Template:Math are required to satisfy
- <math>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})).</math>
Nonassociative examplesEdit
A particularly natural example of a nonassociative medial magma is given by collinear points on elliptic curves. The operation Template:Math for points on the curve, corresponding to drawing a line between x and y and defining Template:Math 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, Template:Math is independent of the choice of a neutral element on the curve, and further satisfies the identities Template:Math. This property is commonly used in purely geometric proofs that elliptic curve addition is associative.