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
- [math]\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})). }[/math]
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.
See also
- Category of medial magmas
Citations
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
- Toyoda, K. (1941), "On axioms of linear functions", Proc. Imp. Acad. Tokyo 17 (7): 221–227, doi:10.3792/pia/1195578751, https://www.jstage.jst.go.jp/article/pjab1912/17/7/17_7_221/_article
- 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
- 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. http://www.karlin.mff.cuni.cz/~jezek/medial/03.jpg.
- 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. 6. Berlin, New York: Springer-Verlag. pp. 197–280. ISBN 978-3-540-54699-3.
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