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Infinite Atomized Semilattices

Fernando Martin-Maroto, Antonio Ricciardo, David Mendez, Gonzalo G. de Polavieja

TL;DR

This work generalizes atomized semilattice theory to infinite structures, showing that every semilattice is atomizable and that the framework supports infinite representations through atomizations and admissible terms. It introduces full crossing, proves its equivalence to principal congruence quotients, and establishes that the freest models can be obtained while preserving the positive theory Th$_0^+$. The paper also develops a robust treatment of redundancy, distinguishing redundancy and weak redundancy, and proves that finitely generated semilattices admit atomizations composed of non-redundant atoms, with implications for completions and freest realizations in the infinite setting. Overall, the results underpin Algebraic Machine Learning with a rigorous infinite-structure theory, enabling representations of richer data and rules via infinite atomizations and controlled quotients.

Abstract

We extend the theory of atomized semilattices to the infinite setting. We show that it is well-defined and that every semilattice is atomizable. We also study atom redundancy, focusing on complete and finitely generated semilattices and show that for finitely generated semilattices, atomizations consisting exclusively of non-redundant atoms always exist.

Infinite Atomized Semilattices

TL;DR

This work generalizes atomized semilattice theory to infinite structures, showing that every semilattice is atomizable and that the framework supports infinite representations through atomizations and admissible terms. It introduces full crossing, proves its equivalence to principal congruence quotients, and establishes that the freest models can be obtained while preserving the positive theory Th. The paper also develops a robust treatment of redundancy, distinguishing redundancy and weak redundancy, and proves that finitely generated semilattices admit atomizations composed of non-redundant atoms, with implications for completions and freest realizations in the infinite setting. Overall, the results underpin Algebraic Machine Learning with a rigorous infinite-structure theory, enabling representations of richer data and rules via infinite atomizations and controlled quotients.

Abstract

We extend the theory of atomized semilattices to the infinite setting. We show that it is well-defined and that every semilattice is atomizable. We also study atom redundancy, focusing on complete and finitely generated semilattices and show that for finitely generated semilattices, atomizations consisting exclusively of non-redundant atoms always exist.
Paper Structure (4 sections, 45 theorems, 10 equations, 2 figures)

This paper contains 4 sections, 45 theorems, 10 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Freedom and full crossing
  4. Atom redundancy in infinite semilattices

Key Result

Proposition 2.12

Assume axiom:AS4. Then,

Figures (2)

  • Figure 1: A representation of the decomposition of the set of atoms $\Omega(M)$ compatible with a semilattice $M$, following Lemma \ref{['lemma:redundant_implies_in_model']}, Proposition \ref{['proposition:redundancy_and_weak_redundancy']} and Proposition \ref{['proposition:compatible_atoms_decomposition']}.
  • Figure 2: A representation of the decomposition of the set of atoms $\Omega(M)$ compatible with a complete semilattice $M$, following Lemma \ref{['lemma:redundant_implies_in_model']}, Proposition \ref{['proposition:redundancy_and_weak_redundancy']} and Proposition \ref{['proposition:compatible_atoms_decomposition']}. In particular, this includes finite semilattices.

Theorems & Definitions (120)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • Remark 2.10
  • ...and 110 more