Sharp Elements and Apartness in Domains

  title={Sharp Elements and Apartness in Domains},
  author={Tom de Jong},
  booktitle={Mathematical Foundations of Programming Semantics},
  • T. D. Jong
  • Published in
    Mathematical Foundations of…
    28 December 2021
  • Mathematics
Working constructively, we study continuous directed complete posets (dcpos) and the Scott topology. Our two primary novelties are a notion of intrinsic apartness and a notion of sharp elements. Being apart is a positive formulation of being unequal, similar to how inhabitedness is a positive formulation of nonemptiness. To exemplify sharpness, we note that a lower real is sharp if and only if it is located. Our first main result is that for a large class of continuous dcpos, the Bridges–V… 



Domain Theory in Constructive and Predicative Univalent Foundations

This work further considers algebraic and continuous dcpos, and construct Scott's $D_\infty$ model of the untyped $\lambda$-calculus, which gives a large, locally small, algebraic dcpo with small directed suprema.

Locatedness and overt sublocales

Exhaustible Sets in Higher-type Computation

A complete description of exhaustible total sets is obtained by developing a computational version of a topological Arzela--Ascoli type characterization of compact subsets of function spaces and shows that they are precisely the computable images of the Cantor space.

Domain theory

bases were introduced in [Smy77] where they are called “R-structures”. Examples of abstract bases are concrete bases of continuous domains, of course, where the relation≺ is the restriction of the

Predicative theories of continuous lattices

The paper introduces strong proximity join-semilattice, a predicative notion of continuous lattice which arises as the Karoubi envelop of the category of algebraic lattices and shows that this structure naturally corresponds to the notion ofContinuous lattice in the predicative pointfree topology.

Constructive Domain Theory as a Branch of Intuitionistic Pointfree Topology

Inductively generated formal topologies

Geometric theories of patch and Lawson topologies

It is shown that the construction of Lawson topologies naturally induces a monad on the category of compact regular formal topologies, which is shown to be isomorphic to the Vietoris monad.

Constructive Domains with Classical Witnesses

A constructive theory of continuous domains from the perspective of program extraction is developed, and a representation theorem is proved that precisely delineates the class of representable continuous functions.

A type-theoretic interpretation of constructive domain theory

It is shown that a theory of domains can be developed within a well-defined fragment of (total) type theory, an important step toward constructing a model of all of partial type theory (type theory extended with general recursion) inside total type theory.