Ein Axiomensystem für Baum-Algebren.

*(German)*Zbl 0532.05058The concept of a tree algebra is derived from the graph-theoretical concept of a tree. If T is a tree, let P be a ternary operation on the vertex set V(T) of T defined so that P(x,y,z) is the (uniquely determined) vertex of T contained simultaneously in a path from x to y, in a path from x to z and in a path from y to z. Then V(T) together with P forms an algebra called a tree algebra; it can be described by certain axioms. Tree algebras can be considered more generally - as algebras fulfilling the mentioned axioms, but not necessarily corresponding to trees. In the paper the so-called discrete tree algebras are studied. A segment on such an algebra is defined by means of P and, conversely, P is defined by means of the segment. A system of three axioms in terms of segments is introduced and its equivalence with another axiom system (in terms of P) for tree algebras is proved.

Reviewer: B.Zelinka