Apartness relation

Abstract: In constructive theories, an apartness relation is often taken as basic and its negation used as equality An apartness relation should be continuous in its arguments, as in the case of computable reals A similar approach can be taken to order relations We shall here study the partial order on open intervals of computable reals Since order on reals is undecidable, there is no simple uniformly applicable lattice meet operation that would always produce non-negative intervals as values We show how to solve this problem by a suitable definition of apartness for intervals We also prove the strong extensionality of the lattice operations, where by strong extensionality of an operation f on elements a, b we mean that apartness of values implies apartness in some of the arguments: f(a, b)≠f(c, d) ⊃a≠c∨b≠dMost approaches to computable reals start from a concrete definition We shall instead represent them by an abstract axiomatically introduced order structure

Abstract: The notion of apartness has recently shown promise as a means of lifting constructive topology from the restrictive context of metric spaces to more general settings. Extending the point-subset apartness axiomatised beforehand, we characterize the constructive meaning of ‘two subsets of a given set lie apart from each other'. We propose axioms for such apartness relations and verify them for the apartness relation associated with an abstract uniform space. Moreover, we relate uniform continuity to strong continuity, the natural concept for mappings between sets endowed with an apartness structure, which says that if the images of two subsets lie apart from each other, then so do the original subsets. Proofs are carried out with intuitionistic logic, and most of them without the principle of countable choice. Mathematics Subject Classification (2000): 54E05, 54E15, 03F60 Quaestiones Mathematicae 25 (2002), 171-190

Abstract: We give a constructive treatment of some basic concepts and results in semigroup theory. Focusing on semigroups equipped with an apartness relation, we give analogues, from the point of view of apartness, of several classical constructions and results, including transitive closure and congruence closure, free semigroups, periodicity, Rees factors, and Green's relations.