Formal methods in the theories of rings and domains
Davide Rinaldi · Electronic Theses of LMU Munich (Ludwig-Maximilians-Universität München) · 2014
In recent years, Hilbert's Programme has been resumed within the framework of constructive mathematics. This undertaking has already shown its feasability for a considerable part of commutative algebra. In particular, point-free methods have been playing a primary role, emerging as the appropriate language for expressing the interplay between real and ideal in mathematics. This dissertation is written within this tradition and has Sambin's notion of formal topology at its core. We start by developing general tools, in order to make this notion more immediate for algebraic application. We revise the Zariski spectrum as an inductively generated basic topology, and we analyse the constructive status of the corresponding principles of spatiality and reducibility. Through a series of examples, we show how the principle of spatiality is recurrent in the mathematical practice. The tools developed before are applied to specific problems in constructive algebra. In particular, we find an elementary characterization of the notion of codimension for ideals of a commutative ring, by means of which a constructive version of Krull's principal ideal theorem can be stated and proved. We prove a formal version of the projective Eisenbud-Evans-Storch theorem. Finally, guided by the algebraic intuition, we present an application in constructive domain theory, by proving a finite version of Kleene-Kreisel density theorem for non-flat information systems.