Distinguished extensions of a lattice-ordered group

Algebra Universalis, 35 (1996) 85-112

0002 5240/96/010085-28501.50 + 0.20/0 9 1996 Birkh~iuser Verlag, Basel

Distinguished extensions of a lattice-ordered group R. N. BALL

Abstract. Any lattice-ordered group (/-group for short) is essentially extended by its lexicographic product with a totally ordered group. That is, an l-homomorphism (i.e., a group and lattice homomorphism) on the extension which is injective on the/-group must be injective on the extension as well. Thus no/-group has a maximal essential extension in the category IGp of/-groups with l-homomorphisms. However, an /-group is a distributive lattice, and so has a maximal essential extension in the category D of distributive lattices with lattice homomorphisms. A distinguished extension of one/-group by another is one which is essential in D. We characterize such extensions, and show that every /-group G has a maximal distinguished extension E(G) which is unique up to an/-isomorphism over G. E(G) contains most other known completions in which G is order dense, and has most/-group completeness properties as a result. Finally, we show that if G is projectable then E(G) is the e-completion of the projectable hull of G.

1. Statement of results An/-group

( G , . , 1, v , /x) carries s t r u c t u r e s o f two different sorts: ( G , . , l) is

a g r o u p a n d (G, v , /~) is a lattice. T h e structures, w h i c h are r e q u i r e d to be c o m p a t i b l e o n l y i n s o f a r as

gl gr > hi r for all r ~ I.

Figure 1. a and b distinguish c from d.

Vol. 35, 1 9 9 6

Distinguished extensions of a lattice-ordered group

89

Figure 2. g threads the needle formed by h~ and h2.

We prove the following two theorems. THEOREM

1.1. The following are equivalent for an l-group H with l-subgroup

G. (t) H is an essential extension of G in D. (2) For each pair hi < h2 m H there is a pair g1 < g2 in G distinguishing hi from

h2. (3) G is an order dense l-subgroup of H having the threading-the-needle property. (4) G is order dense in H, and for all h in H we have

A Ihg '[ : 1. G

If the conditions o f T h e o r e m 1.1 prevail we shall say that H is a distinguished extension of G. A n /-group will be termed distinguished if it has no proper distinguished extensions. THEOREM

1.2. Every l-group G has a distinguished extension E(G) which is

distinguished. E(G) is unique (up to an l-isomorphism over G) with respect to these properties. I f G is representable then E(G) is the e-completion of the projectable hull

of G.

2. Distinguished extensions In this section we prove T h e o r e m 1.1. We explicitly assume the construction o f the maximal essential extension (in the category of distributive lattices and lattice

90

R.N. BALL

ALGEBRA UNIV.

homomorphisms) B(G) of the distributive lattice G found in Section 1 of [7], which we outline briefly here. For a < b and c _< d define (a, b) < (c, d) to mean that a and b distinguish c from d. This relation constitutes a preorder on

{(a,b):a A (c, d> = 0},

and for x ~_ S(G) set x I = (~ {(c, d ) ~ : (c, d ) ~ x}.

Finally let B(g) = {x c S(G) : x =x•177 ordered by inclusion. Then B(G) is a complete Boolean algebra, with

Vx,= r

I

x,

,

Vol. 35, 1 9 9 6

Distinguished extensions of a lattice-orderedgroup

91

and ~ for complementation. Furthermore G embeds canonically in B(G) by means of the association g~-,{(a,b)

:g v a > b } .

It is in this sense that we will consider G to be a sublattice of B(G). A fact of use in the sequal is that for any x ~ B(G) and a < b in G, xva>_b

,~

(a,b)ex,

xAb_ G is essential in D if and only if there is an injection 0 : K ~ H over G. In particular, if K is an essential extension o f G in D then the unique map 0 : K --. B(G) over G is given by the formula kO={(a,b)

:k v a>b}.

We e m b a r k n o w on the p r o o f of T h e o r e m 1.1. L E M M A 2.2. For a 0), since otherwise we may replace each gi by gT. That is because g { and g~ clearly distinguish 1 from h, and since g~ < g2 there is a prime P for which Pgl < Pg2, hence P g > 1, for if P is any prime of H such that P < Pg then Pg2 > Pg~ and so Ph >- Pg2 > P & > P.

Therefore Ph >_ P h g { 1 > Pg2g{ 1 = Pg,

proving h _>g. [] Now we can demonstrate the equivalence of conditions (1) and (3) of Theorem 1.1. Note that the threading-the-needle property can be visualized in A(R) by saying that each bubble formed by a pair in H is pierced by some permutation in G. T H E O R E M 2.5. An l-group extension G < H is distinguished if and only it enjoys the threading-the-needle property and G is order dense in H. Proof. Suppose that H is a distinguished extension of G, and consider hi < h2 in H. Find distinguishing g~ < g2 in G. We claim that when g is taken to be gl, the positive element g 2 g l I lies in the polar displayed in the threading-the-needle condition. For hi /x g2 -< g] implies gag] ~ /x (hlgi1) + =

( g 2 g ~ -1 A

hlg~l) + =

((g2 A

hl)gi-1) + = 1,

and h2 v gl ->g2 implies (h2g{l) + > g 2 g l -~, meaning that g2g~ -1 ~ (h2gV~) +•177Now suppose that G is an order dense/-subgroup of H which satisfies the threading-theneedle property. To prove the extension essential consider hi < h2 in H. First find an element gl e G for which the polar P of the threading-the-needle condition is nontrivial. Because G is order dense in H there is some 1 < d ~ P r~ G such that d < ( h z g ~ l ) +. Let g2 = d g l . Then gz Ph, i.e., Pa > P > Phg-1, from which follows Plhg-l[ = Phg 1 N/ P g h - I < Pa. This proves the claim, which establishes t h a t / ~ G ]hg-l[ = 1. N o w assume that G is order dense in H, and that /~a ]hg = 1 for all 1 < h e H. To prove the extension essential consider hi < h2 in H, and let d = h 2 h l 1. Let us first consider the case in which there is no 1 < x ~ G § satisfying x 2 _< d. In this case find g E G such that [big-l[ ~ d, let

11

X = [ b i g - l [ A d < d,

and let y = d x 1 > 1 . Note that x / x y = l , for otherwise l < x / x y _ < d and (x/x y) 2 _< d violate the case hypothesis. In particular, y v x = y x = d. Then for any prime P omitting y we have x e P, so P d = P ( x v y) > P = P x = PIhlg-ll, and ( h l g - l ) + ~ P. It follows that y ~ (hlg y then

1)+• Again, if P is any prime omitting

P < P y < P ( x v y) = P d = P h 2 h ~ l implies Pg = Phi < Ph2, so (h2g 1) + (~ p. Therefore y lies in the convex/-subgroup

96

R . N . BALL

ALGEBRA UNIV.

generated by (h2g l)+, and in particular

ye(hzg-i)+•177

1)+•162

The remaining case occurs when x 2 < d for some 1 < x < d. The claim is that any g e G such that ]xhlg-ll ~ x satisfies the threading-the-needle condition. F o r consider such a g, let w denote xhlg -1, and let z = (Ixwl 1)+ > 1. There are two subcases, in the first o f which z r (hxg 1)+•177 In this case choose y ~ ( h ~ g - I ) +~ with 1 < y < z. Then for any prime P omitting y it is true that P]w[ < Px, so

Pw -1 u, v s G, meaning u{a, b)v E hO. This insight leads to the notion of the shadow of a set x =_S(G). D E F I N I T I O N . The shadow of a subset x ~_ S(G) is

x ' = {uv : l >_u, v eG, < a , b > e x } •177 In A(N) we have associated with a set x ~_ S(G) the open subset O _~ N2 which is the union of the bubbles formed by the pairs in x. The shadow of x, in turn, is associated with the open subset of the plane consisting of all points southeast of the points of O, i.e., x ' = {(r, s) : r > r 1 and s -_ s, t ~ G then q(x• c_ x • f o r all 1 q, r e G. Proof. S u p p o s e s x t c_x for all 1 > s , t ~ G. I f there is some E x I a n d some 1 _< q, r ~ G such that qr ~ x • then there is some ~ x with qr/x #0. But then /x q - l r i # O, a n d since by a s s u m p t i o n q l r -1 ~ x, this violates the hypothesis that ~ x • [] COROLLARY

3.2. x " = x .for all x ~ S(G).

P r o o f Let y : 0 {sxt : 1 > s, t ~ G}. By L e m m a 3.1 sx't =syi•

~ y'• = x'

for 1 > s, t e G. T h e r e f o r e x " x '~• = x'.

[]

F o r a s u b g r o u p P a n d an element u ~ G we shall use pu to d e n o t e the c o n j u g a t e u-lPu.

Vol. 35, 1 9 9 6

Distinguished extensions of a lattice-ordered group

99

L E M M A 3.3. Suppose x ~_ S(G) satisfies sxt ~_ x f o r all 1 >_s, t ~ G, and suppose that (c, d ) ~ S(G). Then (c, d ) ~ x • if and only if f o r all (a, b ) ~ x, all primes P, and all u, v E G, Pva = Pvb

or

Puc > P u d A Pvb

or

Pu < Pv.

P r o o f Suppose the condition fails. By replacing v by u A v if necessary, we m a y assume u _> v; consequently u - i v ( a , b ) t ~ x, where t = a-~vuc A 1. But then P"u 1vat < P~u-~vbt,

PUc < P"d,

and

Pvat = Puc A Pva < Puc,

hence P"(u 1vat) b}

lies in E(G), and

Proof. If (a, b ) 9 1 then a v 1 >_ b imples uav v uv >_ ubv and u ( a , b ) v 9 1 for Therefore 1 ' = 1 . F u r t h e r m o r e i f l < s ~ G t h e n (s 2 , 1 ) E l , while

l_>u, v 9

(s-l, 1)s=s(s

1, 1 ) = ( 1 ,

s ) 6 1,

showing ls # 1 and s l va 1. Therefore 1 9 N o w consider x 9 E(G), (s, t ) E x, and ( a , b 9 S i n c e ( a , b ) = ( a A b Al, b A1),wemayassumea_u, v e G} •177= (xl :-~x2) •177= X l

ON2.

N o w suppose 1 < s e G is given. Find 0 r (sl, tl ) e xl such that ( $ 1 , tl )s e Xl~, and find 0 :~ (s2, t2) e x2 with (s2, t2)Slltl e X~. If (Se, t2)(~x~ we are done, for there exists 0 r (a, b ) < (s2, t2) with (a, b ) e xl c~ x2 and

(a, b )s~ltl (hlh2)O; this is done by showing that an arbitrary 0 # (a, b ) ~ (hlh2)O cannot be disjoint from all elements of the form (s~, t l ) . ( s 2 , t2) where (si, ti)chiO. First find s~_bhs 1, it follows that (s~, t~ ) e hlO. Then find 0 # (s2, t 2 ) E h20 such that (s2, t2)re(h20) • where r=hs . The claim is that ( s i , t l ) . (s2, t2)/x (a, b ) r To prove the claim consider a prime P of H such that Ps 2 < Pt 2. N o w Ph2 >_Pt2 since h 2 v s 2 > t2, and Ph2 _x)}. It is routine to verify that this set is an /-subgroup of E(G) in which G is join and meet dense. It follows from Theorem 4.18 of [4] and from the uniqueness clause of Theorem 3.13 that r is onto. []

In the following question C(G) designates the closure of G in B(G) (Section 1 of [7]) and G/~ designates the a-completion of G [9]. 2. Is G i~ C(G) ~ E ( G ) ? Certainly Gi~< C(G) c~E(G) by Corollary 2.12 of [9]. Is C ( G ) c ~ E ( G ) an /-subgroup of E(G)? =

REFERENCES [l] ANDERSON, M. and FELL, T., Lattice-Ordered Groups, an Introduction. Reidel Texts in the

Mathematical Sciences. Reidel, Dordrecht, 1988. [2] BALBES,R., Projective and injective distributive lattices. Pacific J. Math. (1967), 405-420. [3] BALBES, R. and DWINGER, P., Distributive Lattices. University of Missouri Press, Columbia, Missouri, 1974. [4] BALL,R. N., Convergenceand Cauchy structures on lattice orderedgroups. Trans. Amer. Math. Soc. 259 (1980), 357 392. [5] BALL,R. N., The distinguished completion of a lattice ordered group. In Algebra Carbondale 1980, Lecture Notes in Mathematics 848, pages 208-217. Springer-Verlag, 1980. [6] BALL, R. N., The generalizedorthocompletion and strongly projectable hull of a lattice orderedgroup. Can. J. Math. 34 (1982), 621-661.

112

R.N. BALL

ALGEBRAUNIV.

[7] BALL, R. N., Distributive Cauchy lattices. Algebra Universalis 18 (1984), 134-174. [8] BALL, R. N., Completions of l-groups. In Lattice Ordered Groups, A. M. W. Glass and W. C. Holland, editors, chapter 7, pages 142 174. Kluwer, Dordrecht Boston-London, 1989. [9] BALL, R. N., The structure of the a-completion of a lattice ordered group. Houston J. Math. 15 (1989),481 515. [10] BALL, R. N. and DAVIS, G., The e-completion of a lattice ordered group. Czechoslovak Math. J. 33 (108)(1983), III 118. [11] BERNAU, S. J., Lateral and Dedekind completions of archimedean lattice groups. J. London Math. Soc. 12 (1976), 320 322. [12] BIGARD, A., KEIMEL, K. and WOLFENSTEIN, S., Groupes et Anneaux Reticules. Lecture Notes in Pure and Applied Mathematics 608. Springer-Verlag, 1977. [13] BIRKHOFF, G., Lattice Theory. Colloquium Publications XXV. American Mathematical Society, Providence, 3rd edition, 1967. [14] FUCHS, L., Partially Ordered Algebraic Systems. Pergamon Press, New York, 1963. [ 15] GLASS, A. M. W., Ordered Permutation Groups. London Mathematical Society Lecture Note Series 55. Cambridge University Press, Cambridge, 1981. [16] GRATZER, G., General Lattice Theory. Academic Press, New York, 1978. [17] HOLLAND, W. C., The lattice-ordered group of automorphisms of an ordered set. Mich. J. Math. 10 (1963), 399-408. Dept. of Math. and Computer Science University of Denver Denver, CO 80208 U.S.A.