On Weak Projectivity in Arithmetic

Abstract

In the 19^th century, non-Euclidean geometries were discovered and studied. In the 20^th century, non-Diophantine arithmetics were discovered and studied. Construction of non-Diophantine arithmetics is based on very general mathematical structures, which are called abstract prearithmetics, as well as on the projectivity relation between abstract prearithmetics. In a similar way, as set theory gives a foundation for mathematics, the theory of abstract prearithmetics provides foundations for the theory of the Diophantine and non-Diophantine arithmetics. In this paper, we study relations between operations in abstract prearithmetics exploring how properties of operations in one prearithmetic impact properties of operations in another prearithmetic. In addition, we explore how to build new prearithmetics from existing ones.