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

09/05/2023

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by   Roland Backhouse, et al.

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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.