An illustration of the power of structure theory (Q1840723)

This article contains a sampling of applications of the theory of compact topological semigroups to diverse branches of mathematics. The theory of compact semigroups with one-sided continuity can be used effectively in Ramsey theory; this is illustrated with a semigroup proof of van der Waerden's theorem on arithmetic progressions. The theory of semigroups with separately continuous multiplication is applied to the study of the decomposition of a linear contraction on a Hilbert space, a result that has useful consequences in ergodic theory. Finally the theory of compact semigroups with jointly continuous multiplication can be applied to the problem of classifying topological spaces that can serve as models for the untyped \(\lambda\)-calculus.