zbMATH — the first resource for mathematics

Directions and foldings on generalized trees. (English) Zbl 0888.68091
Summary: In the last years various extensions of the Bass-Serre theory of group actions on simplicial trees have been the subject of much investigation combining elementary geometric considerations with very sophisticated techniques.
In a series of papers a more general concept of tree, including distributive lattices, $$\Lambda$$-trees where $$\Lambda$$ is a lattice ordered Abelian group, and Tits buildings as special cases, is introduced and investigated, while the dual of the category of these generalized trees is described in a paper of the author [The dual of the category of trees, Preprint Series of the Institute of Mathematics of the Romanian Academy, No. 7 (1992)] using a suitable extension of Stone’s representation theorem for distributive lattices.
The present paper is devoted to the study of two basic operators on the class of generalized trees assigning to a generalized tree $$T$$ the generalized tree $$\text{Dir}(T)$$ of the directions on $$T$$, resp. the generalized tree Fold $$(T)$$ of the foldings of $$T$$. The main result of the paper shows that the two operators above commute, i.e. for any generalized tree $$T$$ the generalized trees Fold $$(\text{Dir}(T))$$ and Dir $$(\text{Fold}(T))$$ are canonically isomorphic. The second basic result of the paper, provides an interpretation of the composite opertor $$\text{Fold}\circ\text{Dir}\cong\text{Dir}\circ\text{Fold}$$ in terms of the so-called quasidirections on generalized trees.

MSC:
 68R10 Graph theory (including graph drawing) in computer science
Keywords:
Bass-Serre theory; generalized tree