Bruck-Toyoda theorem

A binary algebraic structure (A,)(A,\cdot) is medial 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). This law is quite close to commutativity: for example, if (A,)(A,\cdot) has a unit element (is a unital magma), then by Eckmann-Hilton argument, it is an Abelian monoid. Without two-sided unit element, the connection with Abelian groups is somewhat more intricate

(Bruck-Toyoda theorem) A quasigroup (A,)(A,\cdot) is medial iff there is an Abelian group structure (A,+)(A,+) on AA and mutually commuting group automorphisms ϕ,ψ:AA\phi,\psi:A\to A and an element hAh\in A such that

ab=ϕ(a)+ψ(b)+h a\cdot b = \phi(a)+\psi(b)+h

An element aa in a magma (A,)(A,\cdot) is left regular if the left multiplication L aL_a by aa is bijective and right regular if the right multiplication R aR_a is bijective. The Bruck-Toyoda theorem is a direct corollary of a more general statement about (nonunital in general) magmas:

Theorem. A medial magma (A)(A\cdot) has a left regular element ff and a right regular element gg such that also f 2f^2 is left regular and g 2g^2 is right regular iff there exist a commutative semigroup structure (A,+)(A,+) on AA, mutually commuting automorphisms ϕ,ψ:AA\phi,\psi:A\to A and a regular element hh in semigroup (A,+)(A,+) such that

ab=ϕ(a)+ψ(b)+h a\cdot b = \phi(a)+\psi(b)+h
category: algebra

Created on November 2, 2013 at 03:28:03. See the history of this page for a list of all contributions to it.