For the meaning of "medial" in anatomy, see anatomical terms of location. For linguistic senses of the word, see medial (linguistics).

In abstract algebra, a medial algebra is a set with a binary operation which satisfies the identity

• (x . y) . (u . z) = (x . u) . (y . z), or xy.uz=xu.yz

Its importance arises in the concept of an auto magma object and representation (reconstruction) theorems. The identity xy.uz=xu.yz has been variously called medial, abelian, alternation, transposition, bi-commutative, bisymmetric, surcommutative, entropic, etc. (see External links: Historical comments)

For instance, for two endomorphisms f and g, with the usual operation between functions

(f*g)(x)= f(x).g(x)

this says we have again a morphism. There are counterexamples for the converse, but not for the cartesian square of the operation.. In particular this is the only equation with the property.

See also

• Medial category

External links

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