Ray Mines

Profinite Completions and Canonical Extensions of Heyting Algebras

We show that the profinite completions and canonical extensions of bounded distributive lattices and of Boolean algebras coincide. We characterize dual spaces of canonical extensions of bounded distributive lattices and Heyting algebras in terms of Nachbin order-compactifications. We give the dual description of the profinite completion

Scattered, Hausdorff-reducible, and hereditarily irresolvable spaces

\widehat{H}

Quasi-apartness and neighbourhood spaces

of a Heyting algebra H, and characterize the dual space of

Extensions and fixed points of contractive maps in Rn

\widehat{H}

Bounded linear mappings of finite rank

. We also give a necessary and sufficient condition for the profinite completion of H to coincide with its canonical extension, and provide a new criterion for a variety V of Heyting algebras to be finitely generated by showing that V is finitely generated if and only if the profinite completion of every member of V coincides with its canonical extension. From this we obtain a new proof of a well-known theorem that every finitely generated variety of Heyting algebras is canonical.

The polydisk Nullstellensatz

Abstract We show that a topological space is hereditarily irresolvable if and only if it is Hausdorff-reducible. We construct a compact irreducible T 1 -space and a connected Hausdorff space, each of which is strongly irresolvable. Furthermore, we show that the three notions of scattered, Hausdorff-reducible, and hereditarily irresolvable coincide for a large class of spaces, including metric, locally compact Hausdorff, and spectral spaces.

The Priestley Separation Axiom for Scattered Spaces

Abstract We extend the concept of apartness spaces to the concept of quasi-apartness spaces. We show that there is an adjunction between the category of quasi-apartness spaces and the category of neighbourhood spaces, which indicates that quasi-apartness is a more natural concept than apartness. We also show that there is an adjoint equivalence between the category of apartness spaces and the category of Grayson’s separated spaces.

Picard’s theorem

Abstract This paper, which is written within the framework of Bishops constructive mathematics, deals with the construction of the fixed point ξ of a contractive self-map f of R n, and with the rate at which the sequence (fn(x)) converges to ξ for any x in R n. It also discusses contractive extensions of contractive mappings on compact subsets of R n, and almost uniform contractions of complete metric spaces.

Various Continuity Properties in Constructive Analysis

Abstract This paper comprises a constructive investigation of the relationship between compactness, finite rank and located kernel for a bounded linear mapping into a finite-dimensional normed space. The main result is that a bounded linear mapping of a normed space into a finite-dimensional normed space is constructively compact if and only if its kernel is located. Several examples are given which highlight the constructive distinction between a mapping into a finite-dimensional space and a mapping with finite-dimensional range.

Finite Dimensional Algebras

The Nullstellensatz for zeros in a closed polydisk of C n is proved constructively with no appeal to choice axioms.