Hints and tips:
  • You may need to look up some unfamiliar terms in the title or description to get more of a sense of what is involved.
  • If the project has "No" in the "Current Availability:" field, it is already taken or not being offered this academic year but may be available again in future years.
  • The supervisor name links to a contact details webpage so, if you are interested, you can arrange to discuss this project or even propose a related topic of your own.
Level: MSci, MSc
Title: Complete lattices and fixed points
Supervisor:

Prof. Thomas Mueller
Research Area: Algebra
Description:

The following famous result of Tarski and Davis characterises completeness of arbitrary lattices.
Theorem 1. A lattice A is complete if, and only if, every order morphism f: A \rightarrow A has a fixed point.
The project aims at a self-contained presentation of this result and its proof. Relevant definitions and background information (for instance on ordinals) should also be provided in full.
Further Reading:
  • A. C. Davis, A characterisation of complete lattices, Pacific J. Math. 5 (1955) 311–319.
  • A. Tarski, A lattice-theoretical fixpoint theorem and its applications, Pacific J. Math. 5 (1955) 289–309.
Key Modules:

To be confirmed with the supervisor.
Other Information:

Chapters I and II of [1] might be useful as background reading on set theory and the theory of ordinals (any other good book on semi-axiomatic set theory might also do).

  1. J. Dugundji, Topology. Allyn and Bacon, Inc., Boston, 1966.
Current Availability: Yes