Medial

	Email		Print
	Bookmark		Link

Medial

This article is about medial in mathematics. For other uses, see medial (disambiguation)
Medial (disambiguation)

Medial has several meanings* In mathematics, medial is a set with a binary operation satisfying certain properties, see medial.* In anatomy, medial is an adjective describing structures near the midline of an animal, see anatomical terms of location or Human Anatomical Terms#Anatomical directions....
.

In abstract algebra

Abstract algebra

Abstract algebra is the subject area of mathematics that studies algebraic structures, such as group , ring , field , module , vector spaces, and algebra over a field....
, 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 × M ? 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. Binary operations can be accomplished using either a binary function or binary operator....
which satisfies the identity

Identity (mathematics)

In mathematics, the term identity has several different important meanings:*An identity is an equality that remains true regardless of the values of any variables that appear within it, to distinguish it from an Equality which is true under more particular conditions....

, or more simply,

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

Any commutative semigroup

Semigroup

In mathematics, a semigroup is an algebraic structure consisting of a nonempty Set S together with an associative binary operation. In other words, a semigroup is an associative Magma ....
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....
.

Discussion

Ask a question about 'Medial'

Start a new discussion about 'Medial'

Answer questions from other users

Full Discussion Forum

Encyclopedia

In abstract algebra

Abstract algebra

Binary operation

Identity (mathematics)

Semigroup

Identity element

Monoid

In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single, associative binary operation and an identity element....
. An elementary example of a nonassociative 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....
can be constructed as follows: take an abelian group

Abelian group

An abelian group, also called a commutative group, is a group satisfying the requirement that the product of elements does not depend on their order ....
except the group of order 2 (written additively) and define a new operation by x * y = (− x) + (− y).

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: ???? meaning "same" and ???f? meaning "shape"....
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, also known as vertices, and morphisms, also known as arrows or edges, such that when selecting two objects any directed path through the diagram leads 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 a fundamental and abstract way to describe mathematical entities and their relationships. A category is composed of a collection of abstract "objects" of any kind, linked together by a collection of abstract "morphism" of any kind that have a few basic properties ....
with a cartesian product

Cartesian product

In mathematics, the Cartesian product is a direct product of sets. The Cartesian product is named after Ren? Descartes, whose formulation of analytic geometry gave rise to this concept....
. (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 ?: V ? V, and an endomorphism of a group G is a group homomorphism ?: G ? G....
s of a medial magma, then the mapping f.g defined by pointwise multiplication

is itself an endomorphism.

Medial

See also

External links