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Generalized Archimedean Groups 23 GENERALIZEDARCHIMEDEAN GROUPS BY ELIAS ZAKON 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 <x <b, a, bQS, xQA always imply xQS. An interval is referred to as infinite if the number of its elements is infinite. Whereas in [3] the concept of a regularly ordered group was de- fined separately for discretely and densely ordered groups, we now give a unified definition; its equivalence with that given in [3] will be proved later. 0.1. Definition. An ordered group A is said to be regularly ordered, with respect to an integer n>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 dis- crete (regular and dense, respectively). From this definition we immediately derive the following corollary, to be used later. 0.2. 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 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 22 ELIAS ZAKON [April Proof. By A-reguIarity, 5 contains infinitely many elements x which are divisible by k. If x is treated as a variable ranging over all such elements of S, then, as is easily checked, the set of all elements of the form x/k —a is, like- wise, an infinite interval of A. Hence, by r-regularity, it contains an r-divisible element, so that we have x/k = a(mod r) for some xES. It now suffices to set x/k = y, to obtain the required result. Some further preliminary definitions and corollaries are given in §1. 1. Preliminaries. I. As in [3], we define the nth congruence invariant oí an abelian group A, denoted by [n]A or, briefly, by [»], to be the maximum (possibly infinite) number of «-incongruent elements in A.(2) If » is a prime, [n] is called a prime invariant. In the infinite case, we set [«] = », without distinguishing between infinities of different cardinalities^). 1.1. Corollary. // a group A is isomorphic with the additive group of all integers, then [n]A =n, n= 1, 2, 3, • ■ ■. In fact, for every n, the group A splits into exactly n residue classes modulo n; i.e., the quotient group A/nA has order n (cf. footnote 2). II. Let A be an additive abelian group, 5 its subgroup, and »>0 an integer. We shall say that 5 is n-serving in A if, for all x, y ES, the relation x = y(mod n) holds in S whenever it holds in A. If this is true for every n, we simply say that 5 is serving in A (Prüfer, [l]). S is said to be n-basic in A, if every element of A is «-congruent to some element of 5. Again, if this holds for every n, we say that 5 is basic in A. 1.2. Corollary. Let Ao and A (AoÇ^A) be regularly discrete groups with one and the same unit 1. Then A o is basic and serving in A. Proof. Let Ai be the subgroup generated by the unit 1, and let aEA. Clearly, all elements of the form a —k-\ (k= 1, 2, 3, • • • ) constitute an in- finite interval of A which, by «-regularity (n being any positive integer), must contain an «-divisible element, say a —k0-i, so that a = A0-i(mod w), and ko-ÍEAi. Thus every element of A is «-congruent to some element of Ai. Since n is arbitrary here, Ai is basic in A and, similarly, A o is basic in A. Moreover, if aEAo, the congruence a = &0-i(mod «) easily implies that « divides a (in both A0 and A) if and only if « divides the integer A0.It follows that « divides a either in both .40 and A, or in none of these groups. This shows that A0 is serving in A, as asserted. 1.3. Corollary. A subgroup S of a torsion-free abelian group A is serving (*) This maximum number always exists and is unique. In fact, as is readily seen, [n]A coincides with the (finite or infinite) order of the quotient group A/nA, where nA is the sub- group of all elements divisible by n. (') The symbol °o will be treated as a positive number with the usual conventions as to operations and inequalities. In particular, we set logp a> «= » for every p>l. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1961] GENERALIZED ARCHIMEDEAN GROUPS 23 (basic) in A if and only if, for every prime p, S is p-serving (p-basic) in A. (For basic subgroups, this holds also if A is not torsion-free.) Proof. Finite induction easily shows that, whenever 5 is /»-serving, it is also ^-serving (k= 1, 2, 3, • ■ • ). It is also readily seen that, whenever 5 is both »w-serving and «-serving (m and n being relatively prime), then 5 is also (mn) -serving. Similarly for basic subgroups. The required result then is obtained by considering each integer n as factored into prime powers. 1.4. Corollary. Let S be a subgroup of an abelian group A, and n>0 an integer. Then, if S is n-serving in A, we have [n]S^ [n]A. If, however, S is n-basic in A, we have [«]S^ [n]A. For, let [n]S = m^ » ; i.e., 5 contains exactly m elements incongruent modulo n. If S is «-serving, these elements are «-incongruent in A as well, whence [«]ef«= [«]5. If, however, S is «-basic, then, clearly, A cannot contain more «-incongruent elements than S; hence [w]^4 ^ [n]S. 1.5. Corollary. If A is discretely ordered, then [n]A ^w, « = 1, 2, 3, • • In fact, let Ai be the subgroup generated by the unit i of A. Then, as is readily seen, ^4i is serving in A and isomorphic with the ordered group of all integers. By 1.1 and 1.4, we then have «= [m]^4i^ [w]-<4,as asserted. III. A subset S of an abelian group A is said to be n-independent (in A) if, for any finite number m of distinct elements xi, Xt, ■ ■ • , xmQS, and any integers ci, ç2, • • • , qm, the relation 2~1a-i ?•*» —0(mod «) in A always implies g¿ = 0(mod «), i= 1, 2, • ■ ■ , m. S is referred to as maximal ii it is not con- tained in any other «-independent subset of A. The existence of (possibly empty) maximal «-independent subsets, for every n, is easily proved by Zorn's lemma. Also easily derived is the following 1.6. Corollary. In a torsion-free abelian group, every n-independent subset is also nk-independent (k=l, 2, 3, •••)• 1.7. Corollary. If an abelian group A is not divisible(A) by a prime p, then A contains at least one nonempty maximal p-independent subset. Proof. Take any element xQA not divisible by p. Then {x} is clearly a nonempty p-independent subset of A. The rest follows by Zorn's lemma. 1.8. Corollary. Let A be an abelian group, and p a prime. Then all max- imal p-independent subsets of A contain exactly the same (finite or infinite) number of elements. This number (called the pth prime rank of A) equals log, [p]A. Proof. Let 5 be a maximal p-independent subset of A, and let m be the (*) We say that a group A is divisible by an integer n >0, or that n divides A, if all elements of A are »-divisible (in A).
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