Applications of set theory in algebra: The clone lattice

With the Erwin-Schrödinger-Research fellowship of the Austrian Science Fund, I intend to visit Professor Friedrich Wehrung at the Laboratoire de Mathématiques Nicolas Oresme CNRS UMR 6139 in Caen, France, for two years. The goal of this project is to acquire techniques from set theory in order to apply them in the investigation of the clone lattice over an infinite base set. A clone is a set of finitary operations on a base set X which contains all projections and which is closed under function composition. The set of all clones on X forms a complete algebraic lattice, called the clone lattice Cl(X). The investigation of Cl(X), with clones having a fairly natural definition and being central objects in universal algebra, is interesting in itself, but also allows for numerous applications in algebra and theoretical computer science. Whereas its study for a finite base set X involves finitary methods and classical methods from algebra, methods from set theory and logic recently turned out to be more adequate when dealing with an infinite base set. In the past seven years, forcing constructions, infinite combinatorics, descriptive set theory, ultrafilters, and other set theoretical constructions have been successfully utilized in infinite clone theory. With the methods I hope to learn from Wehrung, I intend to tackle several problems of this theory, including: Sublattices, intervals, in particular monoidal intervals, dual atomicity, and dependence of the clone lattice on partition properties of the cardinality of the base set. Professor Wehrung is a leading expert in lattice theory. Most importantly, he has a set theoretical background, and has exploited ideas from set theory numerous times to prove deep results in universal algebra; among his most prominent achievements is his recent solution to Dilworth`s problem. I am confident that joint work with him would very much extend my skills in set theory, lattice theory, and in how to combine these two, and I hope that my visit will moreover result in significant contributions to infinite clone theory.