Vol. 64, No. 1, 1976

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Centralizers of transitive semigroup actions and endomorphisms of trees

Charles Frederick Wells

Vol. 64 (1976), No. 1, 265–271

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.

Mathematical Subject Classification 2000
Primary: 05C25
Received: 18 March 1975
Published: 1 May 1976
Charles Frederick Wells