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Axiomatizing small varieties of periodic l-pregroups

Nikolaos Galatos, Simon Santschi

TL;DR

The paper advances the axiomatization and structural understanding of small, periodic ℓ-pregroup varieties generated by the $n$-periodic family $ extbf{F}_n(\mathbb{Z})$. It establishes that the subvariety $ extbf{V}( extbf{F}_n(\mathbb{Z}))$ is defined inside the LP$_n$-theory by a single equation $x \sigma(y)^n hickapprox \sigma(y)^n x$, with $\sigma(y)=y igwedge y^{[1]} igwedge \cdots igwedge y^{[n-1]}$, and analyzes its finitely generated FSIs as exactly the nontrivial $n$-periodic ℓ-pregroups whose group skeleton is totally ordered. The work also describes how FSIs not ℓ-groups decompose as lexicographic products of a finitely generated totally ordered abelian ℓ-group with $ extbf{F}_k(\mathbb{Z})$ for $kigm| n$, and characterizes the lattice of all finite joins of these varieties as the finite downset lattice of divisors of $n$, yielding a complete lattice-theoretic picture. The combination of wreath-product representations, convex subalgebra theory, and lexicographic constructions provides a robust toolkit for understanding $n$-periodic ℓ-pregroups and their residuated-like logics, with precise axiomatizations and a clear description of the subvariety lattice structure.

Abstract

We provide an axiomatization for the variety generated by the $n$-periodic l-pregroup $\mathbf{F}_n(\mathbb{Z})$, for every $n \in \mathbb{Z}^+$, as well as for all possible joins of such varieties; the finite joins form an ideal in the subvariety lattice of l-pregroups and we describe fully its lattice structure. On the way, we characterize all finitely subdirectly irreducible (FSI) algebras in the variety generated by $\mathbf{F}_n(\mathbb{Z})$ as the $n$-periodic l-pregroups that have a totally ordered group skeleton (and are not trivial). The finitely generated FSIs that are not l-groups are further characterized as lexicographic products of a (finitely generated) totally ordered abelian l-group and $\mathbf{F}_k(\mathbb{Z})$, where $k \mid n$.

Axiomatizing small varieties of periodic l-pregroups

TL;DR

The paper advances the axiomatization and structural understanding of small, periodic ℓ-pregroup varieties generated by the -periodic family . It establishes that the subvariety is defined inside the LP-theory by a single equation , with , and analyzes its finitely generated FSIs as exactly the nontrivial -periodic ℓ-pregroups whose group skeleton is totally ordered. The work also describes how FSIs not ℓ-groups decompose as lexicographic products of a finitely generated totally ordered abelian ℓ-group with for , and characterizes the lattice of all finite joins of these varieties as the finite downset lattice of divisors of , yielding a complete lattice-theoretic picture. The combination of wreath-product representations, convex subalgebra theory, and lexicographic constructions provides a robust toolkit for understanding -periodic ℓ-pregroups and their residuated-like logics, with precise axiomatizations and a clear description of the subvariety lattice structure.

Abstract

We provide an axiomatization for the variety generated by the -periodic l-pregroup , for every , as well as for all possible joins of such varieties; the finite joins form an ideal in the subvariety lattice of l-pregroups and we describe fully its lattice structure. On the way, we characterize all finitely subdirectly irreducible (FSI) algebras in the variety generated by as the -periodic l-pregroups that have a totally ordered group skeleton (and are not trivial). The finitely generated FSIs that are not l-groups are further characterized as lexicographic products of a (finitely generated) totally ordered abelian l-group and , where .

Paper Structure

This paper contains 13 sections, 73 theorems, 65 equations, 5 figures.

Table of Contents

  1. Introduction
  2. The group skeleton and convex (normal) subalgebras
  3. The group skeleton
  4. Convex subalgebras
  5. Convex normal subalgebras
  6. On the structure of Fₙ(ℤ)
  7. Lexicographic products
  8. The structure of n-periodic ℓ-pregroups
  9. Representation and consequences
  10. Local subalgebra
  11. Axiomatization of V(Fₙ(ℤ))
  12. Algebras with totally ordered group skeleton
  13. Axiomatization and subvarieties

Key Result

Lemma 2.1

$\ell$-pregroups are exactly the involutive residuated lattices that satisfy $x+y \approx xy$. Hence, they also satisfy $(xy)^r \approx y^rx^r$, $(xy)^\ell \approx y^\ell x^\ell$, and for all $a,b,c$ the latter is called residuation. Also, multiplication distributes over both joins and meets, and the De Morgan laws hold: $(x\mathbin{\vee} y)^r \approx x^r \mathbin{\wedge} y^r$, $(x\mathbin{\wedge}

Figures (5)

  • Figure 1: An illustration of a function $f$ in $\mathbf F_2(\mathbb{Z})$, of $\sigma_2(f)=f \mathbin{\wedge} f^{\ell \ell}$ (the largest invertible element below $f$) and of $\sigma(f)^{-1}f$ (the associated strictly positive flat element).
  • Figure 2: For $f \in \mathbf F_6(\mathbb{Z})$, we have $f^{[l]}(0) > 0$ iff $l \in \{0,2,3,5\}=P_f$, so $\mathop{\mathrm{\lambda}}\nolimits(f) = \bigwedge_{l\in P_f} f^{[l]}$ is shown in the figure.
  • Figure 3: The points $p$ and $q$ for $\mathop{\mathrm{\lambda}}\nolimits(a)$.
  • Figure 4: The map $a\in \mathrm{F}_6(\mathbb{Z})$ has final periodicity $3$, but we have $a(3) < a(0)+3$. Therefore we can generate the positive flat map $c$ with final periodicity $6$.
  • Figure 5: An element $f$ of $\mathbf F_2(\mathbb{Q} \overrightarrow{\times} \mathbb{Z})$, its global component $\widetilde{f} \in \mathbf F_1(\mathbb{Z})$, and two of its local components $\overline{f}_0, \overline{f}_1 \in \mathbf F_2(\mathbb{Z})$.

Theorems & Definitions (145)

  • Lemma 2.1: GJGJKO
  • Remark 2.2
  • Lemma 2.3: Lemma 2.6(1) of GG2
  • Lemma 2.4
  • proof
  • Remark 2.5
  • Lemma 2.6
  • proof
  • Remark 2.7
  • proof
  • ...and 135 more