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Keywords:
locally finite tree; one-way infinite path; acyclic monounary algebra; tree semilattice
locally finite tree; one-way infinite path; acyclic monounary algebra; tree semilattice
Summary:
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.
References:
[1] R. Halin: Über unendliche Wege in Graphen. Math. Ann. 157 (1964), 125–137. DOI 10.1007/BF01362670 | MR 0170340 | Zbl 0125.11701
[2] L. Nebeský: Algebraic Properties of Trees. Acta Univ. Carol., Philologica Monographia 25, Praha, 1969. MR 0274210
[3] L. Nebeský: A tree as a finite set with a binary operation. Math. Bohem. 125 (2000), 455–458. MR 1802293
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