%0 Journal Article
%T Tangled Closure Algebras
%J Categories and General Algebraic Structures with Applications
%I Shahid Beheshti University
%Z 2345-5853
%A Goldblatt, Robert
%A Hodkinson, Ian
%D 2017
%\ 07/01/2017
%V 7
%N Special Issue on the Occasion of Banaschewski's 90th Birthday (II)
%P 9-31
%! Tangled Closure Algebras
%K Closure algebra
%K tangled closure
%K tangle modality
%K fixed point
%K quasi-order
%K Alexandroff topology
%K dense-in-itself
%K dissectable
%K MacNeille completion
%R
%X 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.
%U http://cgasa.sbu.ac.ir/article_42354_09def3b31ada32383d6d12c9644168af.pdf