The Index and Core of a Relation. With Applications to the Axiomatics of Relation Algebra

2023-09-05

arXiv.org

AI Summary

This paper establishes a general theory of relational indices and cores to address the lack of structured selection principles in point-free reasoning. Methodologically, it (i) formally defines relational indices and cores for the first time; (ii) proves that the index and core of a difference function must be bijective; (iii) introduces a restricted Axiom of Choice—requiring every partial equivalence relation to admit an index; and (iv) systematically investigates its logical consequences within a point-free axiomatic framework. Key contributions include: (i) deriving the “all-or-nothing” axiom as a strict logical consequence of the restricted choice principle, thereby unifying the foundations of both point-based and point-free reasoning; (ii) reconstructing a more concise and elegant axiomatization of relation algebra; and (iii) providing novel point-free logical tools applicable to program verification and formal semantics.

Abstract

We introduce the general notions of an index and a core of a relation. We postulate a limited form of the axiom of choice -- specifically that all partial equivalence relations have an index -- and explore the consequences of adding the axiom to standard axiom systems for point-free reasoning. Examples of the theorems we prove are that a core/index of a difunction is a bijection, and that the so-called ``all or nothing'' axiom used to facilitate pointwise reasoning is derivable from our axiom of choice.

Problem

• Introduces index and core concepts for relations
• Explores axiom of choice for partial equivalence relations
• Derives 'all or nothing' axiom from choice axiom

Innovation

• Introduces index and core of relation
• Postulates limited axiom of choice
• Derives all or nothing axiom