Lattices and Topologies
An introductory course for ESSLLI'08
by Guram Bezhanishvili and Mamuka Jibladze
The aim of this course is to provide the basics of two relatively
new branches of mathematics Lattice Theory and Topology,
which play an important role in developing the algebraic and
topological semantics of non-classical logics. They have their
origins in the works of two famous German mathematicians and close
friends Richard Dedekind (1831 -- 1916) and Georg Cantor
(1845 -- 1918).
Lattices are special partially ordered sets in which all nonempty finite subsets possess suprema and infima.
They encode the algebraic behavior of the entailment relation and such basic logical
connectives as conjunction and disjunction, which results in an adequate
algebraic semantics for a variety of logical systems.
On the other hand, a topology on a set X is a collection of
subsets of X - called open sets - containing ∅, X
and closed under finite intersections and arbitrary unions. A
typical example of a topological space is the real line R with
the topology generated by the open intervals of R. In
topological spaces it makes sense to talk about points being in a
neighborhood of a given point. This provides means to talk about
properties that hold locally at a given point, which opens up a door
for a semantics of non-classical logics called the topological
semantics. In several instances, such as the intuitionistic
logic or large classes of modal logics, the topological
semantics is an extension of the relational semantics, which
is more familiar for the ESSLLI audience.
Lattice theory and topology are closely connected. For one, the collection of
open subsets of a topological space always forms a lattice. Moreover, well-behaved lattices,
such as distributive lattices, can be represented as sublattices of the lattice of
open subsets of a topological space. The logical significance of these representation theorems
lies in the fact that they are essentially equivalent to results about completeness of
various non-classical propositional calculi with respect to the topological semantics.
This course is dedicated to several topological representation theorems for distributive
lattices and related structures. We also discuss how these representation theorems yield
topological completeness results. The course will
consist of five 90 minute lectures. Below we give a short outline of each of the five lectures.
Lecture 1 -
Basics of lattice theory: Partial orders and lattices. Lattices as algebras. Distributive laws, Birkhoff's characterization of distributive lattices. Boolean lattices and Heyting lattices.
Lecture 2 - Representation of distributive lattices:
Join-prime and meet-prime elements. Birkhoff's duality between finite distributive lattices and finite posets.
Prime filters and prime ideals. Representation of distributive lattices.
Lecture 3 - Topology: Topological spaces. Closure and interior. Separation axioms.
Compactness. Compact Hausdorff spaces. Stone spaces.
Lecture 4 - Duality: Priestley duality for distributive lattices. Stone duality for Boolean lattices.
Esakia duality for Heyting lattices.
Lecture 5 - Spectral duality and applications to logic:
Spectral duality.
Distributive lattices in logic.
Relational completeness of IPC and CPC.
Topological completeness of IPC and CPC.
Exercises
In these lectures, because of the lack of time, we had to skip a lot
of detail. We decided to turn the missing parts of the proofs into
exercises. Most of the exercises are easy. But some of them are
rather involved. The ones that require some effort are marked with
an asterisk.
Exercises for Lecture 1
Exercises for Lecture 2
Exercises for Lecture 3
Exercises for Lecture 4
Exercises for Lecture 5
Relevant literature
There is an ample amount of literature available on lattice theory and topology. Below we provide a short selection of sources relevant to this course.
Lattices and order:
For an elementary exposition consult the first few chapters of:
Introduction to Lattices and Order - B. A. Davey and Hilary A. Priestley (second edition, 2002 Cambridge University Press)
A Course in Universal Algebra - Stanley N. Burris and H.P. Sankappanavar:
http://www.math.uwaterloo.ca/~snburris/htdocs/ualg.html
Notes on lattice theory - J. B. Nation:
http://www.math.hawaii.edu/~jb/lat1-6.pdf (Chapters 1-6),
http://www.math.hawaii.edu/~jb/lat7-12.pdf (Chapters 7-12).
The following two books are standard references:
Lattice Theory - Garrett Birkhoff (third edition, 1967 AMS)
General Lattice Theory - George A. Grätzer (second edition, 1998 Birkhäuser)
Topology:
The following three are the standard textbooks on general topology:
General Topology - John L. Kelley (1975 Birkhäuser)
General Topology - Stephen Willard (2004 Courier Dover Publications)
Topology - James R Munkres (second edition, 2000 Prentice Hall)
For online sources consult:
Topology Course Lecture Notes by Aisling McCluskey and Brian McMaster, at the Topology Atlas: http://at.yorku.ca/i/a/a/b/23.htm
Elementary Topology: Math 167 - Lecture Notes by Stefan Waner:
http://people.hofstra.edu/Stefan_Waner/RealWorld/pdfs/Topology.pdf
General Topology - Jesper M. Møller:
http://www.math.ku.dk/~moller/e03/3gt/notes/gtnotes.pdf
What is topology? - Neil Strickland:
http://neil-strickland.staff.shef.ac.uk/Wurble.html
Some basic topological terminology and notation from NoiseFactory Science Archives:
http://noisefactory.co.uk/maths/topology.html#Topological terminology and notation
Duality theory:
An elementary introduction to the Priestley duality can be found in:
Introduction to Lattices and Order - B. A. Davey and Hilary A. Priestley (second edition, 2002 Cambridge University Press)
A more sophisticated and in-depth coverage of duality theory (including the spectral duality) can be found in:
Stone spaces - P. T. Johnstone (1982 Cambridge University Press)
Continuous Lattices and Domains - Gerhard Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. Mislove, D. S. Scott (2003 Cambridge University Press)
Bitopologies:
A nice introduction to bitopological spaces is:
Bitopological Spaces, Compactifications and Completions - Sergio Salbany (1974
University of Cape Town)
The following article is a recent survey on bitopological spaces:
Asymmetry and duality in topology - Ralph Kopperman (Topology and its Applications, Volume 66, Issue 1, 8 September 1995, Pages 1-39)
Completeness theorems for intuitionistic and modal logics:
For topological completeness theorems for intuitionistic and modal logics we refer to:
Intuitionistic logic and modality via topology - Leo Esakia (Annals of Pure and Applied Logic, Volume 127, Issues 1-3, June 2004, Pages 155-170)
Handbook of Spatial Logics, Marco Aiello, Ian Pratt-Hartmann, Johan van Benthem eds. (2007 Springer)