Medial
In abstract algebra, a magma which satisfies the identity
- (x . y) . (u . z) = (x . u) . (y . z)
Its importance come from the concept of an auto magma object and the representation (reconstruction) theorems which originates.
For instance, two endomorphisms, say f and g, with the usual (extension of the) operation between functions (i.e. (f*g)(x)= f(x).g(x)), is again a morphism. There are counterexamples for the converse, but NOT for the cartesian square of the operation!. In particular is the only equation with the property.
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)
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