Tree Algebras Research Articles

Symbolic computation of derivations using labelled trees

We discuss the effective symbolic computation of operators under composition. We analyse data structures consisting of formal linear combinations of rooted labelled trees. We define a multiplication on rooted labelled trees, thereby making the set of these data structures into an associative algebra. We then define an algebra homomorphism from the original algebra of operators into this algebra of trees. The cancellation which occurs when non-commuting operators are expressed in terms of commuting ones occurs naturally when the operators are represented using this data structure. This leads to an algorithm which, for operators which are derivations, speeds up the computation exponentially in the degree of the operator.

Varieties of formal series on trees and Eilenberg's theorem

We establish Eilenberg's theorem in the context of formal series on trees. More precisely, we prove that there is a bijection between varieties of tree series and ( E, M)-varieties of tree algebras, where E and M are appropriate classes of epimorphisms and monomorphisms, respectively, in a suitable category. An application is also given.

Betweenness spaces and tree algebras

Betweenness spaces and tree algebras

Axiomatizing schemes and their behaviors

The syntax and the semantics of flowchart algorithms may be conveniently separated. The syntax of a flowchart algorithm is the underlying flowchart scheme. The semantics is given by an interpretation of each symbol appear ing on the nodes of the scheme. One aspect of this theory is the determination of when two schemes have the same “behavior” under all interpretations. When “behavior” is interpreted in the strongest sense as “performing the same stepwise computations,” the collection of equivalence classes of schemes was shown by Elgot [E-MC] to have the structure of an “iterative theory.” The strong behavior of a scheme is precisely captured by the (labeled, locally ordered, locally finite) tree obtained by “unfolding” the scheme in the familiar way. (When the “behavior” of a scheme F is considered to be the partial input-output function computed by F in each interpretation, certain quotients of the trees are the appropriate interpretations.) The collection of labeled schemes and trees is equipped with the same structure, which is preserved by the map which takes a scheme F to its strong behavior [F]. This structure is simple to describe: starting from the atomic and “base” schemes, using just three operations (composition, tupling, and iteration), any scheme may be constructed. The operation of iteration (or “looping”) on schemes and trees was not defined in all cases in the setup of [E-MC]; for example, consider the scheme 1 -+ 1 consisting of only an exit (and the tree which forms its strong behavior). Thus the schemes and trees form a partial algebra. It was to remedy this situation on the semantic algebras of trees and other iterative theories that the notion of an “iteration” theory was introduced (in [BEW]). Iteration theories are (total) algebras with the same operations as iterative theories. The collection of trees which are strong behaviors of schemes may be given the structure of an iteration theory by completing the iteration operation in an almost arbitrary way. The collection of iteration theories is the class of all mode ls of the set of equations valid in all iteration theories of trees. (See [BEW] for details.)

Representation-Finite Algebras whose Auslander-Reiten Quiver is Planar

Throughout the paper A will denote a finite-dimensional connected and basic /c-algebra of finite representation type. We assume the ground field k to be algebraically closed. By FA we denote the Auslander-Reiten quiver of A, which has as vertices the isomorphism classes of indecomposable left A-modules and in which we put an arrow from the class of M to the class of N whenever there exists an irreducible map from M to N. Algebras whose associated Auslander-Reiten quivers have no oriented cycles have played an important role in representation theory: hereditary algebras, tree algebras, simply connected algebras, etc. In this paper we study some conditions which enable us, in some cases, to decide, in terms of the quiver with relations of A, whether FA has oriented cycles or not. Our conditions give, a constructive characterization of all finite representation type algebras A whose Auslander-Reiten quiver FA is planar, that is, FA has no oriented cycles and every Auslander-Reiten sequence in FA has at most two indecomposable summands in its middle term. We thank Prof. F. Larrion for some suggestions about the presentation of this paper.

Unbounded program memory adds to the expressive power of first-order programming logic

It is proved that no logic of programs with unbounded memory is reducible to a bounded memory programming logic. This is achieved by carefully analyzing firstorder definability in the algebra of finite binary trees.

Algebras of iteration theories

The equational class generated by rational algebraic theories was characterized in Esik, Comput. Linguistics and Comput. Languages XIV (1980), 183–207. Here, this class will be called the class of iteration theories. Also, there is a close connection between Elgot's iterative theories and iteration theories. In this paper we introduce algebras for iteration theories, called iteration algebras. Iteration algebras are natural generalization of regular algebras and they are closely related to iterative algebras as well. It is shown that the absolutely free iteration algebras are the algebras of regular trees.

Iteractive factor algebras and induced metrics

An algebra is said to be iterative if every nontrivial finite system of fixed-point equations has a unique solution. A necessary and sufficient condition for a congruence on the algebra of regular trees to induce an iterative factor algebra is presented and possible induced distance functions are discussed.

Fixed-points and algebras with infinitely long expressions: Part I. Regular algebras

This paper introduces a class of algebras (called the class of regular algebras), in which the algebra of regular trees (unfoldments of monadic program schemes) is an initial algebra. This means that we have for the above-mentioned class “semantics” of monadic program schemes. We show how to treat, in a unified way, such concepts as: monadic and recursive monadic program schemes, regular and context-free languages. On the other hand, the investigation of the properties of regular algebras may be of intrinsic interest, in particular this leads to a very nice generalization of the notion of a polynomial in an algebra. These “new” polynomials, in general, are determined by infinitely long expressions, and existence of such polynomials in the class of regular algebras is closely connected with the property that every finite tuple of algebraic mappings has a least fixed-point which is obtainable as a least upper bound of a denumerable chain of “approximations”.

Tolerances and congruences on tree algebras

Tolerances and congruences on tree algebras