Invariants of inversive 2-structures on groups of labels

摘要

For a finite set D of nodes let E2(D) = {(x,y) x,y G D,x~y}. We define an inversive A2-structure g as a function g: Ei{D) — A into a given group A satisfying the property g(x,y) = g{y,x)~l for all {x,y) e Ei{D). For each function (selector) a;D —> A there is a corresponding inversive A2-structure g" defined by g"(x,y) = a(x) g(x,y) o(y)~l. A function mapping each g into the group A is called an invariant if t}(g°) = n(g) for all g and a. We study the group of free invariants n of inversive A2-structures, where n is defined by a word from the free monoid with involution generated by the set Ej(D). In particular, if A is abelian, the group of free invariants is generated by triangle words of the form.