Centralizers of transitive semigroup actions and endomorphisms of trees
Charles Wells · Pacific Journal of Mathematics · 1976
A tree is locally finite if the interval between any two points is finite.A local isomorphism of a tree with itself is a homomorphism which is an isomorphism when restricted to any interval.Two theorems are proved.One characterizes those locally finite trees which have transitive automorphism groups, and those which have transitive local-isomorphism monoids.The other theorem gives necessary and sufficient conditions for a non-injective transformation to be centralized by a transitive permutation group, and necessary and sufficient conditions for it to be centralized by a transitive transformation semigroup.Also, an example is given of a nonlocally-finite tree with transitive automorphism group.