In
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structur ...
, a medial magma or medial groupoid is a
magma
Magma () is the molten or semi-molten natural material from which all igneous rocks are formed. Magma (sometimes colloquially but incorrectly referred to as ''lava'') is found beneath the surface of the Earth, and evidence of magmatism has also ...
or
groupoid
In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways. A groupoid can be seen as a:
* '' Group'' with a partial fu ...
(that is, a
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
with a
binary operation
In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two.
More specifically, a binary operation ...
) that satisfies the
identity
Identity may refer to:
* Identity document
* Identity (philosophy)
* Identity (social science)
* Identity (mathematics)
Arts and entertainment Film and television
* ''Identity'' (1987 film), an Iranian film
* ''Identity'' (2003 film), an ...
: ,
or more simply,
:
for all , , and , 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.
Any
commutative semigroup
In mathematics, a semigroup is a nonempty set together with an associative binary operation. A special class of semigroups is a class of semigroups satisfying additional properties or conditions. Thus the class of commutative semigroups consists o ...
is a medial magma, and a medial magma has an
identity element
In mathematics, an identity element or neutral element of a binary operation is an element that leaves unchanged every element when the operation is applied. For example, 0 is an identity element of the addition of real numbers. This concept is use ...
if and only if it is a
commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Perhaps most familiar as a pr ...
monoid
In abstract algebra, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being .
Monoids are semigroups with identity ...
. The "only if" direction is the
Eckmann–Hilton argument
In mathematics, the Eckmann–Hilton argument (or Eckmann–Hilton principle or Eckmann–Hilton theorem) is an argument about two unital magma structures on a set where one is a homomorphism for the other. Given this, the structures are the sam ...
. Another class of semigroups forming medial magmas are
normal bands. Medial magmas need not be associative: for any nontrivial
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commu ...
with operation and
integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
s , the new binary operation defined by yields a medial magma that in general is neither associative nor commutative.
Using the
categorical definition of
product, for a magma , one may define the
Cartesian square
In mathematics, specifically set theory, the Cartesian product of two sets and , denoted , is the set of all ordered pairs where is an element of and is an element of . In terms of set-builder notation, that is
A\times B = \.
A table ca ...
magma with the operation
: .
The binary operation of , considered as a mapping from to , maps to , to , and to .
Hence, a magma is medial if and only if its binary operation is a magma
homomorphism
In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...
from to . This can easily be expressed in terms of a
commutative diagram
350px, The commutative diagram used in the proof of the five lemma
In mathematics, and especially in category theory, a commutative diagram is a diagram such that all directed paths in the diagram with the same start and endpoints lead to the s ...
, 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 and are
endomorphism
In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a g ...
s of a medial magma, then the mapping defined by pointwise multiplication
:
is itself an endomorphism. It follows that the set of all endomorphisms of a medial magma is itself a medial magma.
Bruck–Murdoch–Toyoda theorem
The Bruck–Murdoch–Toyoda theorem provides the following characterization of medial
quasigroup
In mathematics, especially in abstract algebra, a quasigroup is an algebraic structure that resembles a group in the sense that " division" is always possible. Quasigroups differ from groups mainly in that the associative and identity element pro ...
s. Given an abelian group and two commuting
automorphisms
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms ...
and of , define an operation on by
: ,
where some fixed element of . It is not hard to prove that 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
In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
to a quasigroup defined from an abelian group in this way. In particular, every medial quasigroup is
isotopic to an abelian group.
The result was obtained independently in 1941 by Murdoch and Toyoda. It was then rediscovered by Bruck in 1944.
Generalizations
The term ''medial'' or (more commonly) ''entropic'' is also used for a generalization to multiple operations. An
algebraic structure
In mathematics, an algebraic structure or algebraic system consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplicatio ...
is an entropic algebra if every two operations satisfy a generalization of the medial identity. Let and be operations of
arity
In logic, mathematics, and computer science, arity () is the number of arguments or operands taken by a function, operation or relation. In mathematics, arity may also be called rank, but this word can have many other meanings. In logic and ...
and , respectively. Then and are required to satisfy
:
Nonassociative examples
A particularly natural example of a nonassociative medial magma is given by collinear points on
elliptic curves
In mathematics, an elliptic curve is a Smoothness, smooth, Projective variety, projective, algebraic curve of Genus of an algebraic curve, genus one, on which there is a specified point . An elliptic curve is defined over a field (mathematics), ...
. The operation for points on the curve, corresponding to drawing a line between x and y and defining 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, is independent of the choice of a neutral element on the curve, and further satisfies the identities . This property is commonly used in purely geometric proofs that elliptic curve addition is associative.
Citations
References
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Non-associative algebra