For a finite set D of nodes let
E2(D)={(x, y)[mid ]
x, y∈D, x≠y}.
We define an inversive
Δ2-structure g as a function
g[ratio ]E2(D)→Δ
into a given group Δ satisfying the property
g(x, y)=
g(y, x)−1 for all
(x, y)∈E2(D).
For each function (selector)
σ[ratio ]D→Δ there is a corresponding inversive
Δ2-structure gσ defined by
gσ(x,
y)=σ(x)·g
(x, y)·σ(y)−1.
A function
η mapping each g into the group Δ is called an invariant
if η(gσ)=η(g) for all g
and σ. We study the group of free invariants η of inversive
Δ2-structures, where η is defined by a word from the free monoid
with involution generated by the set E2(D).
In
particular, if Δ is abelian, the group of free invariants is generated
by triangle words of the form
(x0, x1)(x1,
x2)(x2, x0).