Constructive Galois connections: taming the Galois connection framework for mechanized metatheory

AbstractFor every positive integer n,we introduce and discuss an isotone Galois connection between the sets of paths of lengths n in a simple graph and the closure operators on the (vertex set of the) graph. We consider certain sets of paths in a particular graph on the digital line Z and study the closure operators associated, in the Galois connection discussed, with these sets of paths. We also focus on the closure operators on the digital plane Z2 associated with a special product of the sets of paths considered and show that these closure operators may be used as background structures on the plane for the study of digital images.

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Invariance groups of functions and related Galois connections

Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry ◽

10.1007/s13366-020-00525-4 ◽

2020 ◽

Author(s):

Eszter K. Horváth ◽

Reinhard Pöschel ◽

Sven Reichard

Keyword(s):

Boolean Functions ◽

Group Actions ◽

Galois Connection ◽

General Context ◽

Abstract Group ◽

Galois Connections ◽

Invariance Groups ◽

Galois Closures

Abstract Invariance groups of sets of Boolean functions can be characterized as Galois closures of a suitable Galois connection. We consider such groups in a much more general context using group actions of an abstract group and arbitrary functions instead of Boolean ones. We characterize the Galois closures for both sides of the corresponding Galois connection and apply the results to known group actions.

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Characterizations of M-fuzzifying convex spaces via Galois connections

Journal of Intelligent & Fuzzy Systems ◽

10.3233/jifs-210060 ◽

2021 ◽

pp. 1-11

Author(s):

Shao-Yu Zhang

Keyword(s):

Galois Connection ◽

Galois Connections ◽

Reflective Subcategory ◽

Convex Spaces

This paper introduces a special Galois connection combined with the wedge-below relation. Furthermore, by using this tool, it is shown that the category of M-fuzzifying betweenness spaces and the category of M-fuzzifying convex spaces are isomorphic and the category of arity-2 M-fuzzifying convex spaces can be embedded in the category of M-fuzzifying interval spaces as a reflective subcategory.

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Representation of $L$-fuzzy binary relations via a Galois connection

Tamkang Journal of Mathematics ◽

10.5556/j.tkjm.40.2009.507 ◽

2009 ◽

Vol 40 (3) ◽

pp. 287-305

Author(s):

Nistala V. E. S. Murthy ◽

Peruru G. Prasad

Keyword(s):

Complete Lattice ◽

Galois Connection ◽

Binary Relations ◽

Truth Values ◽

Galois Connections ◽

Fuzzy Point ◽

Point Set

Our aim in this Paper is to establish Galois connections between various types of fuzzy binary relations and fuzzy I-ary relations on a crisp set, that take their truth values in a complete lattice, and same type of crisp binary and I-ary relations on the associated fuzzy-point-set.

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