A small result about graph algebras
Notations.
Let X be a set of variables, and T(X) be the set of finite
terms built from X and the binary symbol "*".
The rooted graph g(t) associated to every t from T(X) has its
vertices in V. It is defined
inductively by:
- if t is a variable (from X) then g(t) is the single-point graph
with vertex t, root t, and no edges.
- if t = p*q, then the graph is obtained as the union of vertices
and edges of g(p) and g(q) augmented by an edge between the roots of
g(p) and g(q). The root of g(t) is the root of g(p).
Obviously g(t) is a connected graph.
Proposition. p == q <=> g(p)=g(q)
Proof (unpublished) obtained in July 94.