Generalized archimedean groups
Elias Zakon · Transactions of the American Mathematical Society · 1961
Introduction.In a previous paper [3], the concept of archimedean group was generalized so as to obtain a larger class of groups that were referred to as regularly ordered.As it was shown in [3], these groups, unlike the archimedean ones, admit a formalization in the lower predicate calculus (LPC) of formal logic; yet they cannot be distinguished from archimedean groups by any properties formalizable in the LPC.Thus they can serve as an LPC-substitute for archimedean groups.This result was, however, based on several theorems stated without proof.It is our aim now to supply the proofs of these theorems.Another objective of this paper consists in giving a unified algebraic approach to regularly ordered groups, as a counterpart to the metamathematical study presented in [3].Throughout this paper, the term "ordered group" stands for "totally ordered (and, hence, torsion-free) additive abelian group other than {O}."Such a group, A, is said to be discretely or densely ordered according as it does, or does not, contain a smallest positive element (called its unit, 1).A subset SÇ.A is called an interval if the relations a 0 (briefly, "n-regular"), ii every infinite interval of A contains at least one (and, hence, infinitely many) elements divisible by «. (') If this holds for every n, we simply say that A is regularly ordered.A is referred to as regularly discrete (regularly dense) if its ordering is both regular and discrete (regular and dense, respectively).From this definition we immediately derive the following corollary, to be used later. Corollary.Let a be an element of a regularly ordered group A, and let k, r be positive integers.Then every infinite interval S of A contains an element x of the form x = ky, with y = a(mod r), yQA.Received by the editors March 2, 1960 and, in revised form, September 27, 1960.(') As usual, we say that an element x is divisible by n (briefly, "n-divisible") in a group A, if A contains an element z with x = nz.Two elements x, yQA are said to be congruent modulo n (briefly, "n-congruent") in A, if x-y is divisible by n.We then write x=y (mod n) in A. The negation of this formula is xf^y (mod n) in A ("x and y are n-incongruent in A").21