• provides a fundamental understanding, which properties (on a general level) are expressible via functions and which are expressible via relations (using certain fragements of first-order logic), and how both aspects are related
• improves the capability to apply abstract Galois theoretic methods for the characterisation of certain closure systems, as well as for the classification of the complexity of corresponding logical decision problems
• gives an insight into the basic toolbox and useful background knowledge for current research themes in general algebra

We shall study the “most basic Galois connection in algebra” and the lattices of its Galois closed sets, (certain) clones and relational clones. In particular we shall provide the link between an internal characterisation of such structures (by means of closure properties) and an external description via the operators of a suitable Galois correspondence. If time permits, we shall focus on problems related to generating systems, minimal and maximal clones, and/or applications of these and related methods to the complexity classification of computational problems in theoretical computer science.

The lecture will take place once a week. In the introductory meeting we agreed to meet every Tuesday at 6 pm.

Appointments for exams are available on an individual basis (for moderate numbers of participants). Contrary to previous indications the exams will only consist of an oral part, no written part is necessary.

You are kindly requested to register for this course via TISS. Should you have missed the deadline, please contact the lecturer in the course.

Szendrei, Ágnes (1986). Clones in Universal Algebra. Séminaire de Mathématiques Supérieures. Vol. 99. Presses de l'Université de Montréal, Montréal, QC. ISBN 2-7606-0770-4

Lau, Dietlinde (2006). Function Algebras on Finite Sets. A basic course on many-valued logic and clone theory. Springer Monographs in Mathematics. Springer, Berlin. doi:10.1007/3-540-36023-9. ISBN 978-3-540-36022-3.

Basic knowledge of Algebra I is helpful, but not a precondition.