Algebraic Approach to Locally Finite Trees with One End

Let T be an infinite locally finite tree. We say that T has exactly one end, if in T any two one-way infinite paths have a common rest (infinite subpath). The paper describes the structure of such trees and tries to formalize it by algebraic means, namely by means of acyclic monounary algebras or tree semilattices. In these algebraic structures the homomorpisms and direct products are considered and investigated with the aim of showing, whether they give algebras with the required properties. At the end some further assertions on the structure of such trees are stated, without the algebraic formalization.