Directions and foldings on generalized trees.

*(English)*Zbl 0888.68091Summary: 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.

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 |