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

Contents 1 Medial magmas 2 Generalizations 3 See also 4 External links

[edit] Medial magmas

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 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 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] Generalizations

The term medial or (more commonly) entropic is also used for a generalization to multiple operations. An algebraic structure is an entropic algebra^[2] 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

[edit] See also

• Medial category

[edit] External links

1. ^ Historical comments J.Jezek and T.Kepka: Medial groupoids Rozpravy CSAV, Rada mat. a prir. ved 93/2 (1983), 93 pp
2. ^ [1]