Congruence preserving functions of Wilke’s tree algebras (original) (raw)
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Abstract. Wilke's tree algebra formalism for characterizing families of tree languages is based on six operations involving letters, binary trees and binary contexts. In this paper a completeness property of these oper-ations is studied. It is claimed that all functions involving letters, ...
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A function on an algebra is congruence preserving if, for any congruence, it maps pairs of congruent elements onto pairs of congruent elements. An algebra is said to be affine complete if every congruence preserving function is a polynomial function. We show that the algebra of (possibly empty) binary trees whose leaves are labeled by letters of an alphabet containing at least one letter, and the free monoid on an alphabet containing at least two letters are affine complete.
Demonstratio Mathematica
A tree language of a fixed type τ is any set of terms of type τ. We consider here a binary operation + n on the set W τ (X n) of all n-ary terms of type τ , which results in semigroup (W τ (X n), + n). We characterize languages which are idempotent with respect to this binary operation, and look at varieties of tree languages containing idempotent languages. We also compare properties of semigroup homomorphisms from (P(W τ (X n)); + n) to (P(W τ (X m)); + m) with properties of homomorphisms between the corresponding absolutely free algebras F τ (X n) and F τ (X m).
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A classification scheme for regular languages or finite semigroups was proposed by Pin through tree hierarchies, a scheme related to the concatenation product, an operation on languages, and to the Schützenberger product, an operation on semigroups. Starting with a variety of finite semigroups (or pseudovariety of semigroups) V, a pseudovariety of semigroups ◊ u (V) is associated to each tree u. In this paper, starting with the congruence γ A generating a locally finite pseudovariety of semigroups V for the finite alphabet A, we construct a congruence ≡ u (γ A) in such a way to generate ◊ u (V) for A. We give partial results on the problem of comparing the congruences ≡ u (γ A) or the pseudovarieties ◊ u (V). We also propose case studies of associating trees to semidirect or two-sided semidirect products of locally finite pseudovarieties.