medial magma
{{for|the triple product|Median algebra}}
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
: {{math|1=(x • y) • (u • v) = (x • u) • (y • v)}},
or more simply,
: {{math|1=xy • uv = xu • yv}}
for all {{math|x}}, {{math|y}}, {{math|u}} and {{math|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.{{sfn|ps=|Ježek|Kepka|1983}}
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.{{sfn|ps=|Yamada|1971}} Medial magmas need not be associative: for any nontrivial abelian group with operation {{math|+}} and integers {{math|m ≠ n}}, the new binary operation defined by {{math|1=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 {{math|M}}, one may define the Cartesian square magma {{math|M × M}} with the operation
: {{math|1=(x, y) • (u, v) = (x • u, y • v)}}.
The binary operation {{math|•}} of {{math|M}}, considered as a mapping from {{math|M × M}} to {{math|M}}, maps {{math|(x, y)}} to {{math|x • y}}, {{math|(u, v)}} to {{math|u • v}}, and {{math|(x • u, y • v) }} to {{math|(x • u) • (y • v) }}.
Hence, a magma {{math|M}} is medial if and only if its binary operation is a magma homomorphism from {{math|M × M}} to {{math|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 {{math|f}} and {{math|g}} are endomorphisms of a medial magma, then the mapping {{math|f • g}} defined by pointwise multiplication
: {{math|1=(f • g)(x) = f(x) • g(x)}}
is itself an endomorphism. It follows that the set {{math|End(M)}} of all endomorphisms of a medial magma {{math|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 {{math|A}} and two commuting automorphisms {{math|φ}} and {{math|ψ}} of {{math|A}}, define an operation {{math|•}} on {{math|A}} by
: {{math|1=x • y = φ(x) + ψ(y) + c}},
where {{math|c}} some fixed element of {{math|A}}. It is not hard to prove that {{math|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.{{sfn|ps=|Kuzʹmin|Shestakov|1995}} In particular, every medial quasigroup is isotopic to an abelian group.
The result was obtained independently in 1941 by Murdoch and Toyoda.{{sfn|ps=|Murdoch|1941}}{{sfn|ps=|Toyoda|1941}} It was then rediscovered by Bruck in 1944.{{sfn|ps=|Bruck|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{{sfn|ps=|Davey|Davis|1985}} if every two operations satisfy a generalization of the medial identity. Let {{math|f}} and {{math|g}} be operations of arity {{math|m}} and {{math|n}}, respectively. Then {{math|f}} and {{math|g}} are required to satisfy
:
Nonassociative examples
A particularly natural example of a nonassociative medial magma is given by collinear points on elliptic curves. The operation {{math|1=x • y = −(x + y)}} for points on the curve, corresponding to drawing a line between x and y and defining {{math|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, {{math|x • y}} is independent of the choice of a neutral element on the curve, and further satisfies the identities {{math|1=x • (x • y) = y}}. This property is commonly used in purely geometric proofs that elliptic curve addition is associative.
Citations
{{reflist}}
References
{{refbegin}}
- {{citation
|last=Murdoch |first=D.C.
|title=Structure of abelian quasi-groups
|journal=Trans. Amer. Math. Soc.
|volume=49 |issue=3 |pages=392–409
|date=May 1941
|jstor=1989940 |doi=10.1090/s0002-9947-1941-0003427-2 |doi-access=free
}}
- {{citation
|last=Toyoda |first=K.
|year=1941
|title=On axioms of linear functions
|journal=Proc. Imp. Acad. Tokyo
|volume=17 |issue=7 |pages=221–227
|url=https://www.jstage.jst.go.jp/article/pjab1912/17/7/17_7_221/_article
|doi=10.3792/pia/1195578751 |doi-access=free
}}
- {{citation
|last=Bruck |first=R.H.
|date=January 1944
|title=Some results in the theory of quasigroups
|journal=Trans. Amer. Math. Soc. |volume=55 |issue=1 |pages=19–52
|jstor=1990138 |doi=10.1090/s0002-9947-1944-0009963-x |doi-access=free
}}
- {{citation
|last=Yamada |first=Miyuki
|year=1971
|title=Note on exclusive semigroups
|journal=Semigroup Forum
|volume=3
|issue=1
|pages=160–167
|doi=10.1007/BF02572956
}}
- {{cite journal
|first1=J. |last1=Ježek
|first2=T. |last2=Kepka
|year=1983
|title=Medial groupoids
|journal=Rozpravy Československé Akad. Věd Řada Mat. Přírod. Věd
|volume=93 |issue=2 |pages=93pp
|url=http://www.karlin.mff.cuni.cz/~jezek/medial/03.jpg
|archive-url=https://web.archive.org/web/20110718093325/http://www.karlin.mff.cuni.cz/~jezek/medial/03.jpg
|archive-date=2011-07-18
}}
- {{cite journal
|last1=Davey |first1=B. A.
|last2=Davis |first2=G.
|year=1985
|title=Tensor products and entropic varieties
|journal=Algebra Universalis
|volume=21 |pages=68–88
|doi=10.1007/BF01187558
}}
- {{cite book
|last1=Kuzʹmin |first1=E. N.
|last2=Shestakov |first2= I. P.
|year=1995
|title=Algebra VI
|location=Berlin, New York |publisher=Springer-Verlag
|series=Encyclopaedia of Mathematical Sciences
|volume=6
|chapter=Non-associative structures
|pages=197–280
|isbn=978-3-540-54699-3
}}
{{refend}}