Tangled closure algebras

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Title: Tangled closure algebras
Author(s): Goldblatt, R
Hodkinson, I
Item Type: Journal Article
Abstract: The tangled closure of a collection of subsets of a topological space is the largest subset in which each member of the collection is dense. This operation models a logical ‘tangle modality’ connective, of significance in finite model theory. Here we study an abstract equational algebraic formulation of the operation which generalises the McKinsey-Tarski theory of closure algebras. We show that any dissectable tangled closure algebra, such as the algebra of subsets of any metric space without isolated points, contains copies of every finite tangled closure algebra. We then exhibit an example of a tangled closure algebra that cannot be embedded into any complete tangled closure algebra, so it has no MacNeille completion and no spatial representation.
Publication Date: 1-Jul-2017
Date of Acceptance: 7-Dec-2016
URI: http://hdl.handle.net/10044/1/43152
ISSN: 2345-5853
Publisher: Shahid Beheshti University
Start Page: 9
End Page: 31
Journal / Book Title: Categories and General Algebraic Structures with Applications
Volume: 7
Issue: 1
Copyright Statement: © Shahid Beheshti University. CC BY creative commons copyright license
Keywords: Science & Technology
Physical Sciences
Mathematics
Closure algebra
tangled closure
tangle modality
fixed point
quasi-order
Alexandroff topology
dense-in-itself
dissectable
MacNeille completion
COMPLETIONS
OPERATORS
LOGIC
math.LO
math.LO
math.GN
03G25, 06E25, 06B23, 54H10
Publication Status: Published
Appears in Collections:Faculty of Engineering
Computing



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