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Congruence based proofs of the recognizability theorems for free many-sorted algebras

Juan Climent Vidal, Enric Cosme Llópez

TL;DR

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Abstract

We generalize several recognizability theorems for free single-sorted algebras to the field of many-sorted algebras and provide, in a uniform way and without using neither regular tree grammars nor tree automata, purely algebraic proofs of them based on the concept of congruence.

Congruence based proofs of the recognizability theorems for free many-sorted algebras

TL;DR

...

Abstract

We generalize several recognizability theorems for free single-sorted algebras to the field of many-sorted algebras and provide, in a uniform way and without using neither regular tree grammars nor tree automata, purely algebraic proofs of them based on the concept of congruence.

Paper Structure

This paper contains 9 sections, 73 theorems, 105 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Recognizable subsets of a free many-sorted algebra
  4. Basic terms
  5. Substitutions
  6. Iterations
  7. Quotients
  8. Tree Homomorphisms
  9. Derivors and recognizability

Key Result

Proposition 2.7

Let $X$ and $Y$ be $S$-sorted sets and $f$ an $S$-sorted mapping from $X$ to $Y^{\wp}$. Then there exists a unique $S$-sorted mapping $f^{\mathfrak{p}}$ from $(\mathrm{Sub}(X_{s}))_{s\in S}$ to $Y^{\wp}$ such that $f^{\mathfrak{p}}$ is completely additive, i.e., for every $s\in S$ and every $\mathca

Theorems & Definitions (209)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark
  • Definition 2.5
  • Remark
  • Definition 2.6
  • Proposition 2.7
  • proof
  • ...and 199 more