TY - JOUR
ID - 42354
TI - Tangled Closure Algebras
JO - Categories and General Algebraic Structures with Applications
JA - CGASA
LA - en
SN - 2345-5853
AU - Goldblatt, Robert
AU - Hodkinson, Ian
AD - School of Mathematics and Statistics, Victoria University of Wellington, New Zealand
AD - Department of Computing, Imperial College London, UK.
Y1 - 2017
PY - 2017
VL - 7
IS - Special Issue on the Occasion of Banaschewski's 90th Birthday (II)
SP - 9
EP - 31
KW - Closure algebra
KW - tangled closure
KW - tangle modality
KW - fixed point
KW - quasi-order
KW - Alexandroff topology
KW - dense-in-itself
KW - dissectable
KW - MacNeille completion
DO -
N2 - 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.
UR - http://cgasa.sbu.ac.ir/article_42354.html
L1 - http://cgasa.sbu.ac.ir/article_42354_09def3b31ada32383d6d12c9644168af.pdf
ER -