BOOLEAN-LINEAR SPACES

We introduce an abstract theory that provides a unified treatment of various structures and spaces occurring in both logic and analysis. The theory, called Boolean-linear spaces, is formulated in terms of partially ordered sets. Examples of Boolean-linear spaces include various function spaces, vector lattices, and lattice ordered groups, as well as linear orderings, Boolean algebras, and certain Boolean-valued models. Every Boolean-linear space has a unique completion and is endowed with a concept of convergence. These concepts subsume various standard examples of completion and convergence. Finally, we study the complete Boolean-linear spaces that possess a spectrum; in many examples, the spectrum is the set of all (positive) real numbers. For these spaces we prove the Spectral Theorem and introduce the Function Calculus. As an application, we show that the vector structure of function spaces is determined by their partial order. © 1990.