Continuous lattices in formal topology

Minimal invariant spaces in formal topology

Journal of Symbolic Logic ◽

10.2307/2275567 ◽

1997 ◽

Vol 62 (3) ◽

pp. 689-698 ◽

Cited By ~ 5

Author(s):

Thierry Coquand

Keyword(s):

Direct Consequence ◽

Arithmetical Progression ◽

Standard Result ◽

Continuous Map ◽

Compact Topological Space ◽

Combinatorial Theorem ◽

Minimal Property ◽

Special Instance ◽

Invariant Spaces ◽

Formal Topology

A standard result in topological dynamics is the existence of minimal subsystem. It is a direct consequence of Zorn's lemma: given a compact topological space X with a map f: X→X, the set of compact non empty subspaces K of X such that f(K) ⊆ K ordered by inclusion is inductive, and hence has minimal elements. It is natural to ask for a point-free (or formal) formulation of this statement. In a previous work [3], we gave such a formulation for a quite special instance of this statement, which is used in proving a purely combinatorial theorem (van de Waerden's theorem on arithmetical progression).In this paper, we extend our analysis to the case where X is a boolean space, that is compact totally disconnected. In such a case, we give a point-free formulation of the existence of a minimal subspace for any continuous map f: X→X. We show that such minimal subspaces can be described as points of a suitable formal topology, and the “existence” of such points become the problem of the consistency of the theory describing a generic point of this space. We show the consistency of this theory by building effectively and algebraically a topological model. As an application, we get a new, purely algebraic proof, of the minimal property of [3]. We show then in detail how this property can be used to give a proof of (a special case of) van der Waerden's theorem on arithmetical progression, that is “similar in structure” to the topological proof [6, 8], but which uses a simple algebraic remark (Proposition 1) instead of Zorn's lemma. A last section tries to place this work in a wider context, as a reformulation of Hilbert's method of introduction/elimination of ideal elements.

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Linear types and approximation

Mathematical Structures in Computer Science ◽

10.1017/s0960129500003200 ◽

2000 ◽

Vol 10 (6) ◽

pp. 719-745 ◽

Cited By ~ 4

Author(s):

MICHAEL HUTH ◽

ACHIM JUNG ◽

KLAUS KEIMEL

Keyword(s):

Full Subcategory ◽

Linear Logic ◽

Domain Theory ◽

Classical Domain ◽

Complete Distributivity ◽

Linear Types ◽

Continuous Lattices

We study continuous lattices with maps that preserve all suprema rather than only directed ones. We introduce the (full) subcategory of FS-lattices, which turns out to be *-autonomous, and in fact maximal with this property. FS-lattices are studied in the presence of distributivity and algebraicity. The theory is extremely rich with numerous connections to classical Domain Theory, complete distributivity, Topology and models of Linear Logic.

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Some operators and dimensions in modular meet-continuous lattices

Journal of Algebra and Its Applications ◽

10.1142/s0219498818500949 ◽

2018 ◽

Vol 17 (05) ◽

pp. 1850094 ◽

Cited By ~ 1

Author(s):

Mauricio Medina Bárcenas ◽

José Ríos Montes ◽

Angel Zaldívar Corichi

Keyword(s):

Complete Lattice ◽

Monotone Function ◽

Continuous Lattice ◽

Continuous Lattices

Given a complete modular meet-continuous lattice [Formula: see text], an inflator on [Formula: see text] is a monotone function [Formula: see text] such that [Formula: see text] for all [Formula: see text]. If [Formula: see text] is the set of all inflators on [Formula: see text], then [Formula: see text] is a complete lattice. Motivated by preradical theory, we introduce two operators, the totalizer and the equalizer. We obtain some properties of these operators and see how they are related to the structure of the lattice [Formula: see text] and with the concept of dimension.

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