A Galois Correspondence for Digital Topology

Šlapal, Josef

doi:10.1007/978-1-4020-1898-5_13

Cited by 9 publications

(5 citation statements)

References 6 publications

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“…There is a natural way of associating a pretopology v ρ on a set S with any reexive binary relation ρ on S: we put X.v ρ = {x ∈ S| ∃y ∈ X : (x, y) ∈ ρ} for every X ⊆ S -see [21]. Let X ⊆ S. If ρ is a reexive and symmetric binary relation on S, then X.v ρ = X.ρ.…”

Section: Remark If In the Denition Of A Grounded Closure Operator Vmentioning

confidence: 99%

Closure operators associated with networks

V{S}Lapal¹,

Pfaltz²

2017

HJMS

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We study network (i.e., undirected simple graph) structures by investigating associated closure operators and the corresponding closed sets. To describe the dynamic behavior of networks, we employ continuous transformations and neighborhood homomorphisms between them. These transformations and homomorphisms are then studied. In particular, the problem of preserving generators by continuous transformations and that of preserving minimal dominating sets by neighborhood homomorphisms are dealt with.

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Section: Remark If In the Denition Of A Grounded Closure Operator Vmentioning

confidence: 99%

Closure operators associated with networks

V{S}Lapal¹,

Pfaltz²

2017

HJMS

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“…A transformation S f −→ S ′ is said to be continuous with respect to an operator, α, or more simply α-continuous, if for all sets Y ∈ S, Y.α.f ⊆ Y.f.α ′ , [1,17,19,30,31]. In the referenced literature, continuity is only considered with respect to a closure operator, ϕ.…”

Section: Continuous Transformationsmentioning

confidence: 99%

Domination and Closure

Pfaltz

2015

Preprint

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An expansive, monotone operator is dominating; if it is also idempotent it is a closure operator. Although they have distinct properties, these two kinds of discrete operators are also intertwined. Every closure is dominating; every dominating operator embodies a closure. Both can be the basis of continuous set transformations. Dominating operators that exhibit categorical pull-back constitute a Galois connection and must be antimatroid closure operators. Applications involving social networks and learning spaces are suggested.

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“…Although this definition of "continuity" has been well established, c.f. [10,35,47], it might merit some further justification. A continuous transformation f in a Euclidean world can be described by "for any set…”

Section: Continuous Transformationsmentioning

confidence: 99%

Mathematical Evolution in Discrete Networks

Pfaltz¹

2014

Math. Appl.

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Abstract. This paper provides a mathematical explanation for the phenomenon of "triadic closure" so often seen in social networks. It appears to be a natural consequence when network change is constrained to be continuous. The concept of chordless cycles in the network's "irreducible spine" is used in the analysis of the network's dynamic behavior.A surprising result is that as networks undergo random, but continuous, perturbations they tend to become more structured and less chaotic.

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A Galois Correspondence for Digital Topology

Cited by 9 publications

References 6 publications

Closure operators associated with networks

Closure operators associated with networks

Domination and Closure

Mathematical Evolution in Discrete Networks

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