A note on “Posets having continuous interval”

doi:10.1016/j.tcs.2020.06.025

Theoretical Computer Science

Volume 837, 12 October 2020, Pages 207-208

https://doi.org/10.1016/j.tcs.2020.06.025 Get rights and content

Abstract

In this note we present a counterexample for a result in [2], where the concept of continuous interval posets (CI-posets for short) was introduced. A CI-poset is a poset in which every nonempty interval $[x, y]$ is a continuous poset (in the restricted order) and the above mentioned result is about continuous functions spaces $[X ⟶ P]$ , where X is a core compact topological space and P is a CI-poset (with its Scott topology).

Section snippets

Counterexample

For all the basic concepts and notations of this note the main reference is [1].

The result for which we will present a counterexample is:

Proposition 1.1

[2]

Let P be a continuous poset such that (i) Each interval of P is a sup-semilattice and (ii) Each principal ideal has a (necessarily unique) smallest element. For a core compact space X, each principal ideal of $[X ⟶ P]$ is a continuous poset and a sup-semilattice with smallest element. If additionally P is a dcpo, then $[X ⟶ P]$ is a continuous dcpo.

In [2], this

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This work was partially financed by funding agency CAPES Brazil - Finance Code 001. (Coordenação de Aperfeiçoamento de Pessoal de Nível Superior).

References (2)

J.D. Lawson et al.
Posets having continuous intervals
Theor. Comp. Sci.
(2004)
G. Gierz et al.
Continuous Lattices and Domains
(2003)

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NoteA note on “Posets having continuous interval”

Abstract

Section snippets

Counterexample

[2]

Declaration of Competing Interest

Acknowledgements

Theor. Comp. Sci.

Continuous Lattices and Domains

Note
A note on “Posets having continuous interval”