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Representability for distributive quasi relation algebras via generalised ordinal sums

Andrew Craig, Claudette Robinson, Wilmari Morton

TL;DR

The paper extends Galatos' generalized ordinal sum construction from residuated lattices to quasi relation algebras (qRAs) and proves that the sum $ ext{K[L]}$ preserves the qRA structure under suitable irreducibility and oddness conditions. It then leverages poset-based relational representations to show that the generalized sum of a finite odd Sugihara chain (notably $ ext{S}_3$) with a representable DqRA is itself representable, and that all finite Sugihara chains are finitely representable. The results yield constructive, finite representations for $ ext{S}_n$ and establish a pathway to representability via generalized sums, with implications for the algebraic understanding of DqRAs and Sugihara chains. Overall, the work advances representability theory for distributive qRAs and provides explicit finite representations for all finite Sugihara chains.

Abstract

We extend the work of Galatos (2004) on generalised ordinal sums of residuated lattices. We show that the generalised ordinal sum of an odd quasi relation algebra (qRA) satisfying certain conditions and an arbitrary qRA is again a qRA. In a recent paper by Craig and Robinson (2024), the notion of representability for distributive quasi relation algebras (DqRAs) was developed. For certain pairs of representable DqRAs, we prove that their generalised ordinal sum is again representable. An important consequence of this result is that finite Sugihara chains are finitely representable.

Representability for distributive quasi relation algebras via generalised ordinal sums

TL;DR

The paper extends Galatos' generalized ordinal sum construction from residuated lattices to quasi relation algebras (qRAs) and proves that the sum preserves the qRA structure under suitable irreducibility and oddness conditions. It then leverages poset-based relational representations to show that the generalized sum of a finite odd Sugihara chain (notably ) with a representable DqRA is itself representable, and that all finite Sugihara chains are finitely representable. The results yield constructive, finite representations for and establish a pathway to representability via generalized sums, with implications for the algebraic understanding of DqRAs and Sugihara chains. Overall, the work advances representability theory for distributive qRAs and provides explicit finite representations for all finite Sugihara chains.

Abstract

We extend the work of Galatos (2004) on generalised ordinal sums of residuated lattices. We show that the generalised ordinal sum of an odd quasi relation algebra (qRA) satisfying certain conditions and an arbitrary qRA is again a qRA. In a recent paper by Craig and Robinson (2024), the notion of representability for distributive quasi relation algebras (DqRAs) was developed. For certain pairs of representable DqRAs, we prove that their generalised ordinal sum is again representable. An important consequence of this result is that finite Sugihara chains are finitely representable.

Paper Structure

This paper contains 7 sections, 25 theorems, 48 equations, 13 figures.

Table of Contents

  1. Preliminaries
  2. Quasi relation algebras
  3. Sugihara chains
  4. Generalised ordinal sums of qRAs
  5. Representable distributive quasi relation algebras
  6. Representable DqRAs via generalised ordinal sums
  7. Finite representations of finite Sugihara chains

Key Result

Lemma 2.1

Let $\mathbf{K}=\langle K, \wedge_{\mathbf{K}}, \vee_{\mathbf{K}}, \cdot_{\mathbf{K}}, \backslash_{\mathbf{K}}, /_{\mathbf{K}}, 1_{\mathbf{K}}\rangle$ be a residuated lattice with $1_{\mathbf{K}}$ irreducible with respect to $\backslash_{\mathbf{K}}$ and $/_{\mathbf{K}}$. Then $\mathbf{K}$ satisfies

Figures (13)

  • Figure 1: Examples of small qRAs.
  • Figure 2: Finite Sugihara chains $\mathbf{S}_2$ to $\mathbf{S}_5$.
  • Figure 3: qRAs $\mathbf{K}_1$, $\mathbf{L}_1$ and their generalised ordinal sum, the qRA $\mathbf{K}_1[\mathbf{L}_1]$.
  • Figure 4: DqRAs $\mathbf{K}_2$, $\mathbf{L}_2$ and their non-distributive generalised ordinal sum $\mathbf{K}_2[\mathbf{L}_2]$.
  • Figure 5: The poset $\mathbf{X}_{\mathbf{S}_2}$ used to represent $\mathbf{S}_2$ (left) and the representation of $\mathbf{S}_2$ (right).
  • ...and 8 more figures

Theorems & Definitions (53)

  • Lemma 2.1
  • proof
  • Lemma 2.2: G04
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • ...and 43 more