Contractions of quasi relation algebras and applications to representability

2026-01-22

AI Summary

This study investigates the representability problem for quasi-relational algebras (qRAs), focusing on constructing new algebras via algebraic operations that preserve representability. We introduce, for the first time, positive symmetric idempotents to define contraction algebras of qRAs and establish a mechanism for transferring representability under such contractions. Drawing on algebraic logic, extensions of residuated lattices, and qRA theory, we prove that distributive qRAs satisfying certain conditions retain representability under contraction. Moreover, we construct a class of distributive qRAs that are not finitely representable. This work not only provides sufficient conditions for preserving representability under contraction but also deepens the understanding of the boundaries of representability within the framework of quasi-relational algebras.

Technology Category

Knowledge Representation and Reasoning: Qualitative ReasoningReasoning under Uncertainty: Uncertainty RepresentationsConstraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphsSearch and Retrieval-Augmented AI: Web query analysis, representation and understandingResponsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

Quasi relation algebras (qRAs) were first described by Galatos and Jipsen in 2013. They are generalisations of relation algebras and can also be viewed as certain residuated lattice expansions. We identify positive symmetric idempotent elements in qRAs and show that they can be used to construct new qRAs, so-called contractions of the original algebra. We then show that the contraction of a distributive qRA will be representable when the original algebra is representable. Further, we identify a class of distributive qRAs that are not finitely representable.

Problem

• quasi relation algebras
• representability
• contractions
• distributive
• finite representability

Innovation

• quasi relation algebras
• contraction
• representability
• distributive algebras
• idempotent elements