M9130 Lattice Theory

Faculty of Science
Autumn 2003
Extent and Intensity
2/0. 2 credit(s) (fasci plus compl plus > 4). Type of Completion: zk (examination).
Teacher(s)
doc. RNDr. Josef Niederle, CSc. (lecturer)
Guaranteed by
doc. RNDr. Josef Niederle, CSc.
Department of Mathematics and Statistics – Departments – Faculty of Science
Contact Person: doc. RNDr. Josef Niederle, CSc.
Prerequisites
M3150 Algebra II && M6100
Basic courses in set theory, discrete mathematics and algebra.
Course Enrolment Limitations
The course is also offered to the students of the fields other than those the course is directly associated with.
fields of study / plans the course is directly associated with
  • Mathematics (programme PřF, M-MA, specialization Discrete Mathematics)
Course objectives
The course is an introduction to lattice theory. Examples of lattices in various fields of mathematics are presented. The course is in particular concentrated on complete lattices as topped intersection structures, algebraic lattices and domains, fixed points, distributivity, relation between Boolean and distributive lattices and topology, complemented modular lattices and projective spaces, ortholattices and Hilbert spaces.
Syllabus
  • Complete lattices: Upsets and downsets, topped intersection structures , closure operators, Dedekind-MacNeille completion, Galois connection, concept lattices
  • Algebraic lattices and domains: Algebraic intersection structures, algebraic closure operators, algebraic lattices, domains
  • Fixed points: Fixed point theorems
  • Distributivity: Continuous lattices, frames
  • Ideals and filters: Prime ideals, maximal ideals
  • Boolean and distributive lattices and topology: Duality between finite distributive lattices and finite ordered sets, representation by lattices of sets, representation of boolean and bounded distributive lattices in dual space, duality
  • Complemented modular lattices and projective spaces
  • Ortholattices and Hilbert spaces
Literature
  • BIRKHOFF, Garrett. Lattice Theory. Third edition. Providence: A. M. S., 1979. info
  • DAVEY, B. A. and H. A. PRIESTLEY. Introduction to Lattices and Order. Cambridge: Cambridge University Press, 1990, 248 pp. Cambridge Mathematical Textbooks. ISBN 0-521-36766-2. info
  • SZÁSZ, Gábor. Einführung in die Verbandstheorie. Budapest: Akadémiai Kiadó, 1962. info
Language of instruction
Czech
Further Comments
The course is taught once in two years.
The course is taught: every week.
The course is also listed under the following terms Autumn 2004, Autumn 2006, Autumn 2009, Autumn 2019.
  • Enrolment Statistics (Autumn 2003, recent)
  • Permalink: https://is.muni.cz/course/sci/autumn2003/M9130