Number systems characterized by their operative properties

Abstract

In our paper Structuring Systems of Natural, Positive Rational and Rational Numbers [The Teaching of Mathematics 22, 1 (2019)], we have studied operative properties of number systems (i.e., the properties of operations and the order relation). In the same paper we have selected a number of operative properties of the system N of natural numbers with 0 which we called the basic operative properties of N. Let {S, +, ·, <} be a structure, where S is a non-empty set, “+”, “·” are two binary operations and “<” is the order relation. We called provisionally such a structure N-structure, when its axioms are basic operative properties of N taken abstractly and we proved that the system N of natural numbers with 0 is the smallest N-structure. Here we rename the N-structure and call it the ordered semifield. Adding to the axioms of the ordered semifield the axiom: (∀a)(∃b) a + b = 0, then such a structure we call the ordered semifield with additive inverse and adding to the same axioms, the axiom: (∀a ≠ 0)(∃b) a · b = 1, we call such a structure the ordered semifield with multiplicative inverse. When both of these axioms are added to the axioms of the ordered semifield, then such a system of axioms coincides with the axioms of the ordered field. In this note we prove that the system of integers is the smallest ordered semifield with additive inverse and that the system of positive rational numbers with 0 is the smallest ordered semifield with multiplicative inverse. The fact that the system of rational numbers is the smallest ordered field is well known. At the end of this note we also include a proof of this fact.