Tangled Closure Algebras

Document Type : Research Paper


1 School of Mathematics and Statistics, Victoria University of Wellington, New Zealand

2 Department of Computing, Imperial College London, UK.


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.


Dedicated to Bernhard Banaschewski on the occasion of his 90th birthday


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