Medial magmas

In abstract algebra

Abstract algebra

Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...

, a medial magma

Magma (algebra)

In abstract algebra, a magma is a basic kind of algebraic structure. Specifically, a magma consists of a set M equipped with a single binary operation M \times M \rightarrow M....

(or medial groupoid) is a set with a binary operation

Binary operation

In mathematics, a binary operation is a calculation involving two operands, in other words, an operation whose arity is two. Examples include the familiar arithmetic operations of addition, subtraction, multiplication and division....

 which satisfies the identity

Identity (mathematics)

In mathematics, the term identity has several different important meanings:*An identity is a relation which is tautologically true. This means that whatever the number or value may be, the answer stays the same. For example, algebraically, this occurs if an equation is satisfied for all values of...

, or more simply,

using the convention that juxtaposition has higher precedence. This
identity has been variously called medial, abelian, alternation, transposition, interchange, bi-commutative, bisymmetric, surcommutative, entropic, etc.

Any commutative semigroup

Semigroup

In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative binary operation. A semigroup generalizes a monoid in that there might not exist an identity element...

 is a medial magma, and a medial magma has an identity element

Identity element

In mathematics, an identity element is a special type of element of a set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them...

 if and only if it is a commutative monoid

Monoid

In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element. Monoids are studied in semigroup theory as they are naturally semigroups with identity. Monoids occur in several branches of mathematics; for...

. Another class of semigroups forming medial magmas are the normal bands. Medial magmas need not be associative: for any nontrivial abelian group

Abelian group

In abstract algebra, 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 their order . Abelian groups generalize the arithmetic of addition of integers...

 and integers m ≠ n, replacing the group operation with the binary operation yields a medial magma which in general is neither associative nor commutative.

A magma M is medial if and only if its binary operation is a homomorphism

Homomorphism

In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures . The word homomorphism comes from the Greek language: ὁμός meaning "same" and μορφή meaning "shape".- Definition :The definition of homomorphism depends on the type of algebraic structure under...

 from the Cartesian square M x M to M. This can easily be expressed in terms of a commutative diagram

Commutative diagram

In mathematics, and especially in category theory, a commutative diagram is a diagram of objects and morphisms such that all directed paths in the diagram with the same start and endpoints lead to the same result by composition...

, and thus leads to the notion of a medial magma object in a category

Category (mathematics)

In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...

 with a cartesian product. (See the discussion in auto magma object

Auto magma object

In mathematics, a magma in a category, or magma object, can be defined in a category with a cartesian product. This is the 'internal' form of definition of a binary operation in a category....

.)

If f and g are endomorphism

Endomorphism

In mathematics, an endomorphism is a morphism from a mathematical object to itself. For example, an endomorphism of a vector space V is a linear map ƒ: V → V, and an endomorphism of a group G is a group homomorphism ƒ: G → G. In general, we can talk about...

s of a medial magma, then the mapping f.g defined by pointwise multiplication


is itself an endomorphism.

Bruck-Toyoda theorem

The Bruck-Toyoda theorem provides the following characterization of medial quasigroup

Quasigroup

In mathematics, especially in abstract algebra, a quasigroup is an algebraic structure resembling a group in the sense that "division" is always possible...

s. Given an abelian group

Abelian group

In abstract algebra, 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 their order . Abelian groups generalize the arithmetic of addition of integers...

 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. In particular, every medial quasigroup is isotopic

Isotopy of loops

- Isotopy of quasigroups :Let and be quasigroups. A quasigroup homotopy from Q to P is a triple of maps from Q to P such that\alpha\circ\beta = \gamma\,for all x, y in Q...

 to an abelian group.

Generalizations

The term medial or (more commonly) entropic is also used for a generalization to multiple operations. An algebraic structure

Algebraic structure

In abstract algebra, an algebraic structure consists of one or more sets, called underlying sets or carriers or sorts, closed under one or more operations, satisfying some axioms. Abstract algebra is primarily the study of algebraic structures and their properties...

is an entropic algebra 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