Medial
From Wikipedia, the free encyclopedia
- This article is about medial in mathematics. For other uses, see medial (disambiguation).
In abstract algebra, a medial magma (or medial groupoid) is a set with a binary operation which satisfies the identity
, 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.[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. An elementary example of a nonassociative medial quasigroup can be constructed as follows: take an abelian group (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 from the Cartesian square M x 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
is itself an endomorphism.
[edit] See also
[edit] External links
- ^ Historical comments J.Jezek and T.Kepka: Medial groupoids Rozpravy CSAV, Rada mat. a prir. ved 93/2 (1983), 93 pp
