nLab mediality

A binary algebraic structure $(A,\cdot)$ satisfies the medial law or mediality if for all $a,b,c,d\in A$

$$(a\cdot b)\cdot(c\cdot d) = (a\cdot c)\cdot (b\cdot d)$$

Medial unital magmas are, by Eckmann-Hilton argument, automatically Abelian monoids. There is a relation between medial quasigroups and Abelian groups, given by Bruck-Toyoda theorem.

Every transpose (conjugate, in the sense of associated ternary relation) conditions to mediality is also the mediality condition. An old term for mediality is also entropy condition/property (“entropic quasigroups”).

• Maciej Niebrzydowski, Józef H Przytycki, Entropic magmas, their homology and related invariants of links and graphs, Algebraic & Geometric Topology 13 (2013) 3223–3243 doi

category: algebra