Some Results in Continuous Logic

We develop C^ω-valued ``continuous'' logic and present some model-theoretic results obtained in this context. First, we consider a description of the Bohr compactification of an abelian topological group (or even a general functional structure) as a type space in this logic. This is described in the context of the dualities that underpin the logic and it is obtained through the use of maps to compact metrizable structures. Next we consider the structure of δ-stable formulas φ(x;y). Partitions of the sorts x, y of a structure are produced such that φ(x;y) is "almost constant" in each pair of parts. Then this result is transferred to a finite context to obtain a continuous version of Malliaris and Shelah's Stable Regularity Lemma. Finally, we consider a double extension of the classical result of positive primitive elimination in modules, first to abelian groups with length functions, and then to abelian groups with length functions and homomorphisms to compact groups. The latter requires a different formalism of "continuous logic", which we flesh out in some detail.