M9130 Lattice Theory

Faculty of Science
Autumn 2019
Extent and Intensity
2/0/0. 2 credit(s) (příf plus uk k 1 zk 2 plus 1 > 4). Type of Completion: zk (examination).
doc. RNDr. Jan Paseka, CSc. (lecturer)
Guaranteed by
doc. RNDr. Jan Paseka, CSc.
Department of Mathematics and Statistics - Departments - Faculty of Science
Supplier department: Department of Mathematics and Statistics - Departments - Faculty of Science
Mon 14:00–15:50 M3,01023
M3150 Algebra II && M1120 Discrete Mathematics
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 deals with more demanding topics from lattice theory and ordered sets. The first theme will be the existence of free objects in categories of semilattices, lattices, distributive lattices, and Boolean algebras, and their complete counterparts - complete semilattice, complete lattices, frames, and complete Boolean algebras. Another topic will be algebraic and continuous ordered sets, including the introduction of Scott's topology. Using duality between coherent frames and distributive lattices through the theory of ideals, we prove Stone's duality between Boolean algebra and Stone's spaces. Finally, we touch the connection between C*-algebras and quantales.
Learning outcomes
At the end of this course, students should be able to: * define more demanding notions of lattice theory and ordered sets; * explain learned theoretical results; * apply learned methods to concrete exercises.
  • Free objects in semilattices and lattices: Semilattice, lattices, distributive lattices, Boolean algebras. Complete semilattices, complete lattices, frames, complete Boolean algebras.
  • Algebraic and continuous ordered sets: Algebraic ordered set, continuous ordered set, Scott's topology.
  • Stone duality: Distributive lattices, coherent frames, and coherent spaces. Boolean algebras and Stone spaces.
  • C*-algebras and quantales: C* -algebras, frames, quantales, Gelfand duality.
  • JOHNSTONE, P. T. Stone spaces. Cambridge: Cambridge University Press, 1982. 370 s. ISBN 0-521-23893-5. info
  • ROSENTHAL, Kimmo I. Quantales and their applications. Essex: Longman Scientific & Technical, 1990. 165 pp. Pitman Research Notes in Mathematics Series 234. ISBN 0582064236. info
  • 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
Teaching methods
Lectures and discussion.
Assessment methods
Lecture with discussions. The exam is oral with written preparation.
Successful passing of the exam presupposes presentation of the selected topic.
Language of instruction
Further comments (probably available only in Czech)
Study Materials
The course can also be completed outside the examination period.
The course is taught only once.
Teacher's information
The lessons are usually in Czech or in English as needed, and the relevant terminology is always given with English equivalents. The target skills of the study include the ability to use the English language passively and actively in their own expertise and also in potential areas of application of mathematics. Assessment in all cases may be in Czech and English, at the student's choice. The course can also be completed outside the examination period.
The course is also listed under the following terms Autumn 2003, Autumn 2004, Autumn 2006, Autumn 2009.
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