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A tree as a finite nonempty set with a binary operation. (English) Zbl 0963.05032

Summary: A (finite) acyclic connected graph is called a tree. Let \(W\) be a finite nonempty set, and let \(\mathbf{H}(W)\) be the set of all trees \(T\) with the property that \(W\) is the vertex set of \(T\). We will find a one-to-one correspondence between \(\mathbf{H}(W)\) and the set of all binary operations on \(W\) which satisfy a certain set of three axioms (stated in this note).

MSC:

05C05 Trees
05C75 Structural characterization of families of graphs
20N02 Sets with a single binary operation (groupoids)
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