From Logic to Discrete Geometry via Lattices

Abstract

The author's journal article "Lattice properties of partial orders for complex matrices via orthogonal projectors" utilized lattice theory, the core of algebraic logic, in order to prove several results on the geometry of matrices. More specifically, it contains a study of the different geometric structures that the intervals of complex square matrices get when sorted by three important partial orders in matrix theory (viz., the left-star order, the star order and the core order). The chapter on order-theoretic lattices contains this paper, as well as a complete and finitely axiomatizable foundation of ultrafinitist mathematics and a connection between the arithmetical hierarchy and the irrationality measure.The author's journal articles "Some properties of the factors of Fermat numbers" and "Some applications of Baaz's generalization method to the study of the factors of Fermat numbers" are part of an ongoing research project on the geometry of numbers, which had its origin in Baaz's paper "Note on the generalization of calculations". The common procedure in these articles is the application of a new technique of extractive proof theory, called Baaz's generalization method, to different proofs of compositeness of some concrete Fermat numbers. The information that was extracted from these proofs led to several new results, among which stands out a geometric characterization of the factors of Fermat numbers in terms of point-lattices and of a new concept called cover. The chapter on number-theoretic lattices contains these papers, as well as further investigation on the theory of covers, related results on the factorization of near-square numbers and of Mersenne numbers, an iterative expression of the products of the first consecutive generalized Fermat numbers and a detailed study of a new object, called Hervás-Contreras chain.