Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-58537Special Issue on the Occasion of Banaschewski's 90th Birthday (II)20170701Tangled Closure Algebras93142354ENRobertGoldblattSchool of Mathematics and Statistics, Victoria University of Wellington, New ZealandIanHodkinsonDepartment of Computing, Imperial College London, UK.Journal Article20160920The 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.http://cgasa.sbu.ac.ir/article_42354_09def3b31ada32383d6d12c9644168af.pdf