Above and below subgroups of a lattice-ordered group

In an /-group G, this paper defines an /-subgroup A to be above an /-subgroup B (or B to be below A) if for every integer n, a e A, and b s B, n(\a\ A |/>|) < |a|. It is shown that for every /-subgroup A, there exists an /-subgroup B maximal below A which is closed, convex, and, if the /-group G is normal-valued, unique, and that for every /-subgroup B there exists an /-subgroup A maximal above ß which is saturated: if 0 = x A y and x + y s A, then x e A. Given /-groups A and B, it is shown that every lattice ordering of the splitting extension G = A X B, which extends the lattice orders of A and B and makes A he above B, is uniquely determined by a lattice homomorphism it from the lattice of principal convex /-subgroups of A into the cardinal summands of B. This extension is sufficiently general to encompass the cardinal sum of two /-groups, the lex extension of an /-group by an o-group, and the permutation wreath product of two /-groups. Finally, a characterization is given of those abelian /-groups G that split off below: whenever G is a convex /-subgroup of an /-group H, H is then a splitting extension of G by A for any /-subgroup A maximal above G in H. 0. Introduction. Though the groups are not in general abelian, the notation is additive with the lattice operations taking priority: a + b r\ c is s + (t A c), -b A c is (-b) V (-c), but na A b is (na) A b. g+, g~, and \g\ denote as usual g V 0, -g V 0, and g V (-g), respectively, g is a component of h if \g\ A \h g\ = 0; we remind the reader that components commute and that x + y = x V y whenever x and y are disjoint, that is, whenever x A y = 0. <ë(G), #/((?), Jf (G), if (G), and @(G) denote respectively the lattices of convex /-subgroups, principal convex /-subgroups, closed convex /-subgroups, cardinal summands, and polars. (The reader unfamiliar with these basic concepts can find them in [6].) í?(G) is a complete Boolean algebra with ¡f(G) as a subalgebra. We write A < B to indicate that A is an /-subgroup of G, and G(a) for (g e G: \g\ < n\a\ for some integer n}, the principal convex /-subgroup of G generated by a. If A is any subset of G we use (A) to designate the /-subgroup generated by A. Note that the elements of (A) have the form Vf., Af=fX¡j, where the x^'s are from the subgroup generated by A. Finally, b <§: a means that n\b\ < \a\ for all integers n. Received by the editors October 23, 1984 and, in revised form, August 8, 1985. 1980 Mathematics Subject Classification (1985 Revision). Primary 06F15; Secondary 20E22. 'This work was done when the first author was a visiting associate professor of mathematics at the University of Kansas. He would like to express his appreciation for the hospitality he enjoyed there. ©1986 American Mathematical Society 0002-9947/86 $1.00 + $.25 per page 1 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 2 R. N. BALL, PAUL CONRAD AND MICHAEL DARNEL A subgroup P of G is regular if it is maximal in ^(G) with respect to omitting some element g, and we say that P is a value of g in this case. Such a subgroup is prime inasmuch as it contains at least one element from each disjoint pair. A set T of regular subgroups is plenary provided that fir = 0 and that P G T and P ç Q for a regular subgroup Q imply Q G T. Such a set forms a root system in the inclusion order; that is, a partially ordered set such that for any y g T the set {S g T: 8 > y} is a chain. Most often we view T as an index set and use small Greek letters to indicate its elements; this abuse of notation allows us to write the corresponding regular prime as Gy for y g T and to use Gy = C\{GS: S g T and Gy c Gs} for its cover. Note that Gy is prime but not in general regular, and so C need not correspond to a member of T. G is said to be normal-valued if each Gy is normal in Gy. If B is any subset of G, then T(B) designates {y g T: Gy is a value of some fcefi}. An element g is special if T(g) is a singleton, and in this case we term its value special also. Given /-groups A and B their cardinal sum A ffl B is their group direct sum with cardinal order: a + b > 0 if and only if a > 0 and Z> > 0. If (Gx: X G A} is a set of /-groups, then their cardinal product (it should be called the large sum but is not) is TIGX = {/: A -* UGX: /(A) g Gx for all X g A} with componentwise group and lattice operations. The cardinal sum (or small sum) is {/ G T1GX: {X G A: /(X) # 0} is finite}. Now suppose T is any root system and S any subgroup of the real numbers R. V(T,S) designates {/: A -» S: supp(/) admits no infinite ascending chains}, where supp(/) ={XgA:/(X)tí=0}. The group operations in V(T, S) are componentwise and the order is determined by declaring / > 0 if and only if /(X) > 0 for all maximal elements X of supp(/). The /-groups V(T, S) are universal for abelian /-groups [12]. 2(r, S) = {g g V(T, S): supp(g) is finite}. Now let 38 be a Boolean algebra. lm will denote the greatest element of 38 and 0m the least element. If 38 = ¿P(G) for an /-group G, then, la= G and 03 = 0. 1. Above and below subgroups. In this section we characterize in various ways what it means for an /-subgroup A to lie above another /-subgroup B in an /-group G. We show that for any /-subgroup A there is an /-subgroup B maximal below A which is convex, closed, and unique in the normal valued case, and that for any /-subgroup B there is an /-subgroup A maximal above B which is closed and saturated. These results are fundamental for the rest of the paper. Proposition 1.1. The following are equivalent for elements a and b in G. (a) \a\ A \b\ « |fl|. (b) n\b\ A |a| is a component ofn\b\ for all positive integers n. (c) There is no prime P and no integer n for which P < P + \a\ < P + nb. If these conditions hold, then G(b) = (G(b) n G(a))tB (G(b) n ax), and we designate the projection l-homomorphisms va: G(b) -» G (a) and pa: G(b) -» ax ; that is, xva G G(a), xpa G a x , and x = xva + xpa for all x g G(b). In particular, xv a = x + A a + + jc + A a x~ A a + x~A a~ for all x g G(b). Proof. Suppose 0 < a A b « a and let b0 = b A a and bf = b — b(l — (ba)+. We claim b0 A bf = 0, for which it is sufficient to show bx A a = 0. For if P is any License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use above and below subgroups of a lattice-ordered group 3 prime such that P + bx A a > P, then P + b>P + a>P, whence P + a + b ^ P + 2a > P + a and P + 2b > P + b + a ^ P + b > P + a, meaning P + la A (a + b) A (b + a) A lb = P + 2(a A b) > P + a, contradicting a A b «: a. The claim shows that b0 and bx are complementary components of b, hence nb0 and w¿>, are complementary components of nb for any positive integer n. But nbY g a x , hence nb A a = nb0 A a = nb0 since b0 •« a. This shows that (a) implies (b). Now suppose (b) holds and that P is a prime for which P < P + |4 Since n\b\ = n\b\ A \a\ + z for some ze^çP, P + nb < P + n\b\ = P + n\b\ A \a\ < P + \a\, which establishes (c). Finally suppose \a\ A \b\ •« \a\ fails—say n(\a\ A \b\) £ \a\ for some positive integer n. Then there must be some prime P for which P + n(\a\ A \b\) > P + \a\. Observe that P < P +\a\A\b\^ P +\a\< P + n(\a\A\b|) ^ P + n\b\= P + nb V(-n)b. This proves that (c) implies (a). Consider arbitrary a and b satisfying the conditions, and fix x g G(b). Then there is some integer n for which |jc| < n\b\, and the argument that (a) implies (b) above shows that k(x+A \a\) V k(x~A]a\) < A:(|jc| a|«|) < k(n[b\ A \a\) = kn(\b\ A [a]) < \a\ for all positive integers k. That is, the properties above hold when b is replaced by either x+ or x~. Therefore there are unique elements p and q in a1 such that x + = x + A \a\ + p = x + A a + + x + A a~+ p and x~= x~A \a\ + q = x~A a + + x~ A a~+ q. By substituting these expressions into x = x+— x~ one obtains x = xva + (p + q)<a(G(b)nG(a))m(G(b)na±). D If a and b satisfy the conditions of Proposition 1.1 we shall say that a lies above b or that b lies below a. More generally, we shall say that a subset A çz G lies above a subset B ç G, or that B lies below A, if each element of A lies above each element of B. Proposition 1.2. Suppose G is a normal-valued l-group having positive elements a and b and plenary set A of regular primes. Then b «: a if and only if a G Gs for all 8 G A(Z>), and b lies below a if and only ifb g Gs for all 8 g A(a). Proof. If b •« a and 8 G A(Z>), then Gs + nb < Gs + a for all positive integers n, hence a <£ {g G G: Gs + \g\ < Gs + nb for some n) = Gs. Conversely, if it is not true that b •« a then there is some positive integer n and 8 g A with Gs + nb > Gs + a. Clearly a lies in the cover of the unique value of b containing Gs. Now suppose b is below a and consider Í e A(a). Then a g Gs implies a A b g Gs, so Gs = G + a A b = G + b, meaning b G Gs. If b is not below a then there is some 8 g A(a A b) for which a G Gs. But then a <£ Gs implies 8 g A(a), and b € Gs. D The failure of Proposition 1.2 in A(R), the /-group of order-preserving automorphisms of the real numbers R, shows that the hypothesis of normal-valuedness cannot be omitted from Proposition 1.2, or indeed from the other results of this section in which it appears. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 4 R. N. BALL, PAUL CONRAD AND MICHAEL DARNEL Proposition 1.3. For each positive element a in the normal valued ¡-group G, D = {b: \b\ « a} is the unique largest subgroup of G such that D < a, and D G Jt(G). Furthermore, C = a1 B3Z) = C\{GS: 8 g A(a)} = {b: b lies below a) is the unique largest subgroup of G which lies below a, and C G Jf (G). Proof. Let A be any plenary set of regular primes of G. Consider 0 < Z>,,/>2 g D and 8 g A(¿>, + b2). Since A(/>, + b2) Q A(/>,) + A(/>2), a <£ Gs by Proposition 1.2, hence 6, + b2 «: a. It follows that D g 'tf(G), and is therefore clearly maximal with respect to D < a. To show D closed suppose VE = e, where E