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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 $m \neq n$ , 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 $M \times M$ with the operation

(x, y) • (u, v) = (x • u, y • v)

.

The binary operation $•$ of $M$ , considered as a mapping from $M \times M$ 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 $M \times M$ 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–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.^[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

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.

References

Murdoch, D.C. (May 1941), "Structure of abelian quasi-groups", Trans. Amer. Math. Soc., 49 (3): 392–409, doi:10.1090/s0002-9947-1941-0003427-2, JSTOR 1989940
Toyoda, K. (1941), "On axioms of linear functions", Proc. Imp. Acad. Tokyo, 17 (7): 221–227, doi:10.3792/pia/1195578751
Bruck, R.H. (January 1944), "Some results in the theory of quasigroups", Trans. Amer. Math. Soc., 55 (1): 19–52, doi:10.1090/s0002-9947-1944-0009963-x, JSTOR 1990138
Yamada, Miyuki (1971), "Note on exclusive semigroups", Semigroup Forum, 3 (1): 160–167, doi:10.1007/BF02572956
Ježek, J.; Kepka, T. (1983). "Medial groupoids". Rozpravy Československé Akad. Věd Řada Mat. Přírod. Věd. 93 (2): 93pp. Archived from the original on 2011-07-18.
Davey, B. A.; Davis, G. (1985). "Tensor products and entropic varieties". Algebra Universalis. 21: 68–88. doi:10.1007/BF01187558.
Kuzʹmin, E. N.; Shestakov, I. P. (1995). "Non-associative structures". Algebra VI. Encyclopaedia of Mathematical Sciences. Vol. 6. Berlin, New York: Springer-Verlag. pp. 197–280. ISBN 978-3-540-54699-3.