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

(ab)(cd)=(ac)(bd) (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

Last revised on June 22, 2017 at 18:02:53. See the history of this page for a list of all contributions to it.