Archimedean lattices

ARCHIMEDEAN LATTICES

J. M A R T I N E Z

247

Abstract

An archimedean lattice is a complete algebraic lattice L with the property that for each compact element c~L, the meet of all the maximal elements in the interval [0, c] is 0. L is hyper-archimedean if it is archimedean and for each x~L, [x, 1] is archimedean. The structure of these lattices is analysed from the point of view of theil meet-irreducible elements. If the lattices are also Brouwer, then the existence of complements for the compact elements characterizes a particular class of hyper-archimedean lattices.

The lattice of/-ideals of an archimedean lattice ordered group is archimedean, and that of a hyper-archimedean lattice ordered group is hyper-archimedean. In the hyper-archimedean case those arising as lattices of/-ideals are fully characterized.

Notation and terminology

Our set theoret ic no ta t ion is as fol lows: if A and B are subsets o f a set X then

(A c B) A ___ B denotes (proper) con ta inment o f A in B; A\B is the complement o r b in A.

Our lat t ice theoret ic te rminology is s tandard , except where expressly noted that it

is not. The te rminology f rom the theory o f latt ice ordered groups is for the most pa r t

that o f C o n r a d [6].

1. The basic definitions and preliminary remarks

Dur ing most o f our discussion we will be deal ing with an a lgebra ic lat t ice L : an

e lement c of a lat t ice L is compact if whenever c < V i~ r xl (provided the indica ted jo in

exists) then there exist indices il, i2, ..., i, EI such th/tt a<xi~ v x~ .... v x ~ . A n algebraic lattice is then a complete latt ice in which every e lement is the jo in of compac t

elements. I f L is an a lgebraic lattice, c(L) denotes the subset of c ompa c t elements;

clearly c(L) is closed under finite joins.

There are several wel l -known charac ter iza t ions of a lgebraic lattices. We summa-

rize them in one theorem below, fol lowed by exp lana to ry notes and some historical

comments .

1.1. T H E O R E M . Let L be a complete lattice; then the following statements con-

cerning L are equivalent. (a) L is algebraic; (b) L is isomorphic to the lattice of ideals of a join semilattice; (c) L is isomorphic to an algebraic closure system;

Presented by L. Fuchs. Received February 13, 1973. Accepted for publication in final form August 6, 1973.

248 J. MARTINEZ ALGEBRA UNIV.

(d) L is isomorphic to the lattice of O-subalgebras of an O-algebra, over a suitable operator domain O.

1.2. Comments and clarifications 1) A join semilattice is a partially ordered set (henceforth: p.o. set) in which each

pair (a, b} has a join a v b; an ideal of a join semilattice is a subset closed under finite joins, and containing along with each element a, all elements x < a.

2) An algebraic closure system is a subset B of the power set 2 x of a set X', containing the set X', closed under arbitrary intersections, and with the additional property that B is closed under unions of chains (or upward directed families) of subsets in B.B is then a complete lattice in which arbitrary intersections agree with those in 2 x, but joins rarely do.

3) For the definition of an O-algebra and the necessary terminology of universal algebra we refer the reader to [5] and [9]. Some of our preliminary results are known, and can be easily proven in the language of universal algebra; but we shall refrain from introducing its rather elaborate vocabulary in this paper. All our results will flow from lattice theoretic arguments.

4) Various authors contributed to the proof of theorem 1.1. The notion of compactness is due to L. Nachbin [12]; Birkhoff and Frink proved the equivalence of d) and c) [3]; Nachbin [12] showed that a) implies b), while it is easy to see that b) implies a). It is also quite easy to show that b) implies c) and that c) implies a). For further elaboration we suggest that the reader consult [10].

On occasion it will be convenient to view compact elements in an algebraic lattice from a different point of view: call an element a of a lattice join-inaccessible if whenever a = Vi~t xl, with {x~ [ i~I} upward directed, then a=x~o, for some io~I. A compact element is always join-inaccessible, and in an algebraic lattice the converse is true; see [2].

Therefore if L is an algebraic lattice and cec(L), 'there is a subset of elements {m~ I ieI} such that each m~<c, and i f x < c then x < m j for s o m e j s L We shall refer to these elements as being maximal below c.

One further comment on the relationship between algebraic lattices and closure systems. I f such a lattice L is realized as the lattice of subalgebras of some algebra, then the compact elements correspond to the finitely generated subalgebras; see [2].

A lattice L is called a Brouwer lattice if for each pair a, b eL, the set {x~L [ a A x<_ b} has a largest element. If L is complete then it is Brouwer if and only if aA (Va b~) = Vz (a A bx) holds for all a, b ~ L (2~a) ; (see [2].) In particular Brouwer lattices are distributive. In a Brouwer lattice L with least element 0, there is a natural polarity: for each a~L, let a' be the largest element in (x~L [ x A a = 0 } . We get that 1) a<_a", 2)a<b-- .b '<a' and3) a=a' "; i.e. themapa~a' isanauto-Galois-connect iononL. We call aeL a polar if a=a" the subset of polars is a closure system, and forms a

Vol. 3, 1973 ARCHIMEDEAN LATTICES 249

Boolean algebra under inclusion, called the Boolean algebra of polars associated with L. If L is complete then so is the Boolean algebra of polars, since arbitrary meets agree in the two lattices. This algebra is a sublattice of L if and only if a' v a" = 1 for all aeL. (For references regarding this paragraph the reader is urged to consult [2], O. Frink [8] and J. Varlet [13].)

We now arrive at the first crucial and new definition of this paper. Let L be an algebraic lattice; we say that L is archimedean if for each compact element ceL the intersection of the elements which are maximal below c is 0. The motivation for this notion comes from the theory of archimedean lattice ordered groups (henceforth: /-groups): an/ -group G is archimedean if for each pair O<a, beG, nazgb, for some natural number n. It is well known that if G is an archimedean /-group then every principal /-ideal is a subdirect product of copies of additive subgroups of the real numbers, each equipped with the usual ordering, and the converse of this is trivial. (We should observe that this fact seems to be one of those well established bits of knowledge that are never recorded, least of all proved anywhere.) Thus, treating such a principal/-ideal as an/-group in its own right, the intersection of its maximal/-ideals is zero (and these are/-ideals of G). In view of the fact that the compact elements of the lattice L (G) of/-ideals of G are the principal/-ideals, we have that if G is an archimedean/-group then L (G) is an archimedean lattice; the converse is trivial. We remark that the lattice of/-ideals of any/-group is an algebraic, Brouwer lattice; see [6]. The conclusion concerning this special lattice theoretic property of L(G), when G is archimedean, which has led to our definition of an archimedean lattice, first appeared in [4] and is due to Roger Bleier.

On further crucial definition; an algebraic lattice L is hyper-archimedean if L is archimedean and for each xeL the interval ['x, 1] is archimedean; (it is always algebraic!) Again this notion is motivated by/-groups: an/-group G is hyper-archimedean if it is archimedean and every/-homomorphic image is also archimedean. It is im- mediate from the definition that G is hyper-archimedean if and only i f L (G) is hyper- archimedean. (We should point out that here as in the previous paragraph we are taking the groups to be abelian, for convenience in motivating our definitions.)

Before stating our first results, the reader is reminded of one basic definition: an element t < 1 in a lattice L is meet-irreducible if t = A ~ a x~ implies that t=x~ for some /~A. The notion of finite meet irreducibility is defined in the obvious manner.

Below, let L be an arbitrary algebraic lattice; the proofs of the next three lemmas are easy. The first two can be proved by methods in universal algebra; for complete- ness we shall indicate their lattice-theoretic proofs.

1.3. LEMMA. I f x< 1 in L then there is a meet-irreducible element t>_x. Further x is the meet of meet-irreducible elements.

Proof. Since x < 1 there's a compact element c:~x. Using Zorn's lemma and the

250 J. MARTINEZ ALGEBP.A trNIV.

characterization of compactness in terms of join-inaccessibility, one can find an element t > x which is maximal with respect to not exceeding c. One easily checks that t is meet- irreducible.

Suppose {t~ [ i~I} is the family of meet-irreducible elements > x , and y = A t~>x. We can find a compact element d such that d < y but de~x. As before, we find a meet irreducible t o > x, maximal with respect to not exceeding d; yet since d < y we get that d<to, a contradiction. Thus A t i=x.

1.4. LEMMA. The meet of all the meet-irreducibles of L is O. Proof. Directly from 1.3.

1.5. LEMMA. L is Brouwer if and only if it is (finitely) distributive. We leave the proof as an exercise to the reader. We do point out one thing: it is

a routine matter in lattice theory to show that one finite distributive law (globally) implies the other. When dealing with Brouwer lattices we shall have occasion to use the law: a v (b A C)= ( a v b) A (a v c).

1.6. T H E O R E M . Suppose L is a hyper-archimedean lattice; then the subset of meet-irreducibles in L is trivially ordered. Conversely, if L is modular and the set of meet-irreducibles is trivially ordered then L is hyper-archimedean.

Proof. Suppose first that L is hyper-archimedean. A meet-irreducible element t (in any complete lattice) always has a cover f: namely the meet of all the elements that properly exceed t. We will show that here f = 1, for each meet-irreducible t. Since L is hyper-archimedean It, 1] is an Archimedean lattice; then in It, 1] [is the unique atom. I f f < 1, then there is a compact element eEL so that c ~ f; but then c v f is compact in I-t, 1] and c v f > t'. The meet of the maximal elements of It, 1] then exceeds :, contra- dicting the archimedeaneity of It, 1]. Hence each f = 1, and so the meet-irreducibles form a trivially ordered set.

Conversely, suppose L is a modular, algebraic lattice and the set of meet-irreducibles is trivially ordered. Then each one is maximal, and by lemma 1.4 their meet is 0. Index the set of meet-irreducibles by {t~ [ 2~A}, and select c~c(L), with c>0 . There is a 2~A such that c z~ tx; let A' be the subset of all indices )~ for which t~ ~ c. By modularity each such e A t~ is maximal below c, and A ~ ~ A' (t~ A C) = A ~ a t~ = 0; since c was arbitrary it follows that L is archimedean.

Now if xaL , consider I-x, 1"1; the set is the complete set of meet-irreducibles of I-x, 1], and their meet is x; so by the previous paragraph Ix, 1] is archimedean, and therefore L is hyper-archimedean.

We have already seen that in an algebraic lattice, an element p which is maximal with respect to not exceeding some compact element is meet-irreducible. Conversely, i fp is meet-irreducible it is easy to see that it is maximal with respect to not exceeding

Vol. 3, 1973 ARCFIIMEDEAN LATTICES 251

some compac t element. I f c~c(L) and p is maximal with respect to not exceeding c, we say p is a value of c.

1.7. P R O P O S I T I O N . Let L be an archimedean lattice, and 0 < c < d be compact elements; then c and d have a value in common. Conversely, let L be a modular algebraic lattice in which any two comparable compact elements > 0 have a common value; then L is archimedean.

Proof. Suppose L is archimedean and 0 < c < d , with c, d~c(L). O<c~E0, d] , so there is an m, maximal in [0, d] such that c ~ m. By Zorn ' s l emma pick y > m so that it is a value of c; then d s y. I f z > y then z >__ c; also z > m, and hence z > c v m = d. This says that y is also a value of d.

Conversely, suppose L is modular , c, d~c(L) with 0 < c < d ; suppose p is a value of bo th c and d. c is compac t in [-0, d] , and by modular i ty d ^ p is maximal below d, with c s d ^ p . This is enough to show L is archimedean.

1.8. E X A M P L E S . a) I f E is any vector space, the lattice V ( E ) ofsubspaces of E is a hyper-archimedean, modula r lattice, Of course, it is well known that V ( E ) is modular , and the only meet-irreducibles are the subspaces of co-dimension 1.

In fact, if R is a semi-simple Artinian ring, then one can show (see for example [I 1)] that each R-module has a hyper-archimedean lattice of R-submodules .

b) The lattice pictured below is a non-modu la r archimedean lattice.

1

0

c) We can even find a hyper-archimedean non-modu la r lattice:

1

o

d) Consider the lattice

1

0


The meet-irreducibles form a trivially ordered set, yet the lattice is not even archimedean, indicating the importance of modularity in theorem 1.6. This example satisfies the condition of proposition 1.7 as welI, again showing that modularity is indispensable there.

2. Archimedean and hyper-archimedean, Brouwer lattices

We study very carefully here the structure of archimedean, and especially hyper- archimedean Brouwer lattices. In view of the comments regarding the presence of both finite distributive laws, following lemma 1.5, the finite meet irreducibles in a Brouwer lattice are precisely the primes (in a lattice L p ~ L is prime if p _ a ^ b implies that p >__ a or p _ b). This fact will be used frequently.

We record one general result concerning Brouwer lattices; the reader is urged to compare it with theorem 2.2 in [6]. Let us call an element 0 < s of a Brouwer lattice basic if 0 is prime in [0, s]. First a technical preliminary is in order.

2.1. LEMMA. Let L be a Brouwer lattice with O; O<s~L. Then the following are equivalen t:

i) s is basic; ii) s" is a basic;

iii) if O < x < s then x '=s ' ; iv) s' is prime;

v) s' is a minimal prime; vi) s' is a maximal polar;

vii) s" is a minimal polar; viii) s" is a maximal basic element.

Proof. It all depends on the following observation, the proof of which we leave to the reader: i l L is a Brouwer lattice (with 0) and O<a~L then the map p ~ p ^ a is a one to one correspondence between the primes of L that do not exceed a, and the primes of [0, a]. The inverse map assigns to a prime q in [0, a] the largest element p such that p ^ a = q.

i) ~ v) Since 0 is the smallest prime of [0, s] s' is a minimal prime, by the previous paragraph. Trivially v) ~ iv).

iv)-*ii) I f s ' is prime in L then O=S"AS' is prime in [0, s"]. ii) ~ vi) Suppose b is a polar and 1 >b>s ' ; then b'<s" and 0<b ' . Thus b' is basic,

and therefore b=b" is a minimal prime, (since we already know that i) implies v)!) But s' is also prime, and hence b=s'.

It is clear that vi) and vii) are equivalent, so we show that vi) implies iii). If 0 < x_< s then s'<<_x'< 1, and hence by vi) s '=x ' .

iii)--*i) Suppose O < x , y < s and x A y = 0 ; then y < x ' = s ' , which implies that y A s = 0; this is a contradiction.

Finally viii) clearly implies ii), and if ii) holds, while s" <b with b basic, then b" is basic and so by iii) b ' = s ' ; this says that s"= b", a contradiction.

A Brouwer lattice (with 0) L is said to have a basis if there is a maximal set of pairwise disjoint elements {sz] 2~A}, each of which is basic.

Vol. 3, 1973 ARCFIIMEDEAN LATTICES 253

2.2. T H E O R E M . Let L be a Brouwer lattice with O; the following are equivalent. 1) L has a basis; 2) Each element 0 < x c L exceeds a basic element. 3) The Boolean algebra of polars of L is atomic. Proof. 1) ~ 2) Suppose (sx I 2cA} is a basis for L, and 0 < x eL. By the maximality

of the s~, XAs~>O for some p e A ; clearly XAS~ is basic. 2) ~ 3) I f b is a polar, then b_> s for some basic element; it follows that b >_ s", and

by lemma 2.1 s" is an a tom in the Boolean algebra of polars. 3 ) ~ 2 ) Suppose x > 0 , then x">__s, where s is a polar, basic eIement. Clearly

x ^ s > 0 and so it is basic. 2) ~ 1) Let (s~ [ 2cA} be a maximal set of pairwise disjoint basic elements. We

show that it is maximal as a set of pairwise disjoint elements. I f y A s ~ = 0 for each 2cA, and 0 < s < y with s basic, then s ^ s a = O for all ).cA. By the maximality of the s~ relative to basic elements, this is impossible.

We return now to algebraic lattices; in fact, for the remainder of this section L will always represent an algebraic, Brouwer lattice. We point out for the benefit of the readers familiar with the theory of/-groups, the significant fact that most of our results which are analogues of those in that theory, do not require the rather strong condition which is satisfied by lattices of/-ideals (at least in the abelian case): in a lattice of /-ideals the set of primes is a root system: no two incomparable elements have a common lower bound.

2.3. PROPOSITION. L is archimedean if and only if c' = A {all values of c}, for each compact element coL.

Proof. Suppose L is archimedean, and O<ccc(L) ; let {p~ [2cA} be the set of values of c; since c ^ c ' = 0 and pz is prime, px >_ c', for each 2cA. I f c ' < A pz there is a compact element d < A pa so that d~_c', i.e. d ^ c>0 . Since L is archimedean there is an element m maximal below c such that d ^ c:~m. I f y is the largest element of L such that y ^ c = m then y v c covers y, and y is a value of c; so y=p~, for some peA. But then d<p~ and hence d ^ c<p~ ^ c=m, a contradiction. Thus c ' = A p~.

On the other hand let us assume that the indicated condition holds for each O<ccc(L) . Select such a compact element c and let {px [ 2cA} once again be its set of values. By the remark at the beginning of the proof of 2.1 the elements c ^p~ are all the elements maximal below c. Now A z (c ^ p~) = c ^ ( A ~ Pz) = c ^ c' = 0. Thus L is archimedean, and we are done.

I t is tempting in many of our arguments in this section to think that c(L,) should be closed under finite intersections. Later we shall see an example of a hyper-archimedean, Brouwer lattice in which this property fails. But we shall need it soon, so let us say that an algebraic lattice L has thefinite intersection property (FIP) on compact


elements if c (L) is closed under finite intersections. I f an algebraic, Brouwer lattice L has the property that c v c ' = 1 for all compact elements eeL, we say that L has the compact splitting property (CSP).

2.4. T H E O R E M . Let L be given; the following are equivalent. a) L has the CSP. b) L has the FIP, and the set of primes of L is trivially ordered.

In particular, with either of these conditions, L is hyper-archimedean. Proof. a ) ~ b ) Suppose c, dec(L) and c a d = V ~ l x i , where the xl are upward

directed. Now l = d v d ' , so c = ( c ^ d ) v ( c ^ d ' ) , and hence c = V , , , ( x , v ( c ^ d ' ) ) . But then c=X~oV(CAd' ) for a suitable index i0; we conclude that c^d=(x~oV v (c A d ' ) ) A d = (xio A d) v 0 = Xlo. Thus c A d is join-inaccessible, and hence compact; so L satisfies the FIP.

Next suppose p < q are primes, and select a compact element c:gp yet with c < q. Since CA C'=0, c'<p, and so 1 = c v c ' = q v p = q < 1 ; this is a contradiction. It follows that no two primes are comparable.

For the converse of theorem 2.4 we need a lemma; assume L has the FIP.

2.5. LEMMA. There is a one to one correspondence between mblimal primes of L and ultrafilters of c (L). 1) This correspondence is given as follows; if p is a minimal prime, let U ( p ) = {c~c(L) [ c~_p}," its inverse assigns to an ultrafilter M of c(L) the element V {c'tc~M}.

Proof. Let us suppose M is an ultrafilter in c(L), a n d p = V {c' ] c~M}. We show that N ( p ) = M . I f dr and d<p then by compactness d<e'~ v c ; v .--vc;,, with ci~M (l <i<<.k). C=cl AeZA"" ACkSM and c'l v c'2 v . . . v c'k <_e', so d<_c' which means that CA d- -0 ; hence d~M. Conversely, if dzgp then dA c > 0 for all c~M, and since M is an ultrafilter d~M. Hence N ( p ) = M , and we get immediately that p is a prime. I f q is prime and q<<_p then N (q)~_N ( p ) = M , and since N (q) is a filter, N ( q ) = M , so that q=p. The conclusion is that p is a 'minimal prime.

Conversely, suppose p is a minimal prime, and let N = N (p); let M ~ N be an ultrafilter (which exists by Zorn's lemma). By the previous paragraph, q = V {c' [ c~M} is a minimal prime, and N (q )=M. Thus q=p since p is minimal and thence N=M. So N (p) is an ultrafilter, and as before p = V {c'[c~_p, c~c(L)}.

2.5.1. COROLLARY. I f L has the FIP, then p~L is a minimal prime if and only if p= V {c' [ c~p , cec(L)}. I f p is a minimal prime and p>_d~c(L), then p~:d ' .

Now let us prove that b) implies a) in theorem 2.4: suppose cec(L) yet c v c ' < 1. Let p be a meet irreducible so that p >_ c v c'; by assumption, p is a minimal prime and so by corollary 2.5.1 p>_c~p~c ' , a contradiction.

This completes the proof of theorem 2.4.

1) Filter here means proper filter; an ultrafilter is a maximal filter.


In section 4 we shall see that the pair of requirements in b) of 2.4 are irredundant; we give examples where one holds but the other fails. Both of these lattices will be hyper-archimedean, so in particular the fact that the meet irreducibles are trivially ordered does not imply the same thing for the prime elements.

The next theorem is a curious analogue of a result about archimedean/-groups.

2.6. THEOREM. Suppose L is an archimedean, Brouwer lattice, and x ~ L is a polar. Then Ix, 1] is arehimedean.

Proof. First we show this: if L has the property that the meet of its maximal elements {m~[2~A} is 0, and x = x " ~ L then I-x, 1] has the same property. Let A ' = {)~A [ m ~ x } ; if 2~A' then m~>x', whereas y = A {m~ [ 26A '}>x . On the other hand

0 = A { m x [ 2 ~ A } = ( A { m x [ 2 ~ A ' } ) A ( A { m z [ 2 ~ A ' } ) > - - x ' A y ,

implying that y<_x"=x. Thus x=y , and this is precisely what we had in mind for

Ex, l]. Now let L be archimedean, and x = x " e L ; write 1 = V cx, as the join of all compact

elements of L. Then x = V (cx ̂ x), ex ̂ x is a polar in [0, c~] and [-0, c~] has the property of the previous paragraph. Thus ['cx ̂ x, ex] has the property that the meet of the maximal elements is cx ̂ x.

Now suppose d > x is compact in I-x, 1]; it is easy to show that d = x v e for some compact element ceL. Now [c A x, c] ~--Ex, d], and so the meet of the maximal elements below d over x is x. It follows that Ix, I] is archimedean, as desired.

Finally, we have the following isolated results.

2.7. PROPOSITION. Suppose L satisfies the FIP and c' v c" = 1, for each c~c(L). Then if p and q are distinct minimal primes, p v q = 1.

Proof. Suppose p and q are distinct minimal primes, and let a~c(L) so that a < p but azgq. By corollary 2.5.1 a'zgp, and since p is prime a"<p; on the other hand a'<_q. So 1 = a " v a ' < _ p v q , i.e. p v q = l .

The converse of this is false; examples can be obtained from the theory of/-groups. If L does have the property that the prime elements form a root system, then

O<s~L is basic if and only if [-0, s] is a chain. For if s is basic then s' is a prime (lemma 2.1); since the primes exceeding s' are totally ordered so is Is', 1]. Hence [0, s] ~- Is', s' v s] is a chain as well. If [0, s] is a chain then clearly s is basic. If L is also archimedean then it is evident that a basic element is an atom; we have therefore almost proved:

2.8. PROPOSITION. Let L be an archimedean, Brouwer lattice in which the prime elements form a root system; then s v s '= I i f s > 0 is basic.

256 J. MARTINEZ ALGEBRA tsNiv.

Proof. I f s v s ' < 1, pick a compact element c ~ s v s'. I f s $ c then s n c = 0 and so

c < s' < s v s', a contradiction. Hence s < c: from our discussion preceding this proposition it is evident that s ' is the only value of s. By 1.7 it must be a value of c as well; but then c<s v s', which is again absurd.

3. Realizations of hyper-archimedean, Brouwer lattices, as lattices of/-ideals

There are two questions we should as a matter of course address ourselves here. The first one is this: since our notions of an archimedean lattice comes from the theory of/-groups, we should be interested in finding which archimedean, Brouwer lattices

arise as the lattice of/-ideals of an archimedean/-group. The second question concerns the more special hyper-archimedean, Brouwer lattices: if a hyper-archimedean, Brouwer lattice fails to have the CSP, and we shall see that it can, how can we correct

this deficiency by way of embedding it in a reasonable manner in one which does have the CSP? These two questions are related as we will exhibit shortly.

As the reader might guess, the first problem is predictably hard in general terms; we shall solve it in the case of hyper-archimedean, Brouwer lattices.

With regard to the second question, there is a rather obvious embedding which

ought to work and does to a degree. If L is any algebraic, Brouwer lattice with the FIP, then L is isomorphic to the lattice of ideals of the distributive lattice c(L); this is Nachbin's result again. Now the lattice of ideals of a distributive lattice can be

embedded in the lattice of congruences on the same lattice as follows: let D be a distributive lattice (with 0, for convenience), and J be an ideal of D; define a congruence Qj by: aojb if and only if a v d=b v d for some d~J. It is well known that ~o s

is a congruence, and the map J ~ Os is an order embedding; that is, J1 ---J2 if and only if Qj, -~QJ2- It is quite easy to show that the map is in fact lattice preserving.

I f L is now a hyper-archimedean, Brouwer lattice with the FIP, then it can be embedded as a sublattice in the lattice L* of congruences on c(L). By a well-known result in lattice theory (see [2]) such a lattice of congruences is Brouwer, and every

prime element is maximal. Thus L* is a hyper-archimedean lattice which satisfies half of condition b) in theorem 2.4; as far as the author knows it might fail to have the FIP, and this is the main drawback of the embedding.

We summarize the above as follows:

3.1. PROPOSITION. I f L is a hyper-archimedean Brouwer lattice with the FIP, then L can be embedded in a hyper-archimedean] Brouwer lattice in which the primes are trivially ordered. (Note: the reader will of course observe that the embedding just described works for

any algebraic, Brouwer lattice. In particular then, a sublattice of a hyper-archimedean lattice need not be even archimedean.)


Let us now turn to the problem of'realization' of archimedean, Brouwer lattices. I f G is an / -group then L (G), the lattice of/-ideals, is an algebraic, Brouwer lattice with the FIP in which the primes form a root system (see [6]); if G is archimedean, so is L(G), and if G is hyper-archimedean then L(G) has the CSP ([6], theorem 2.4). Our main theorem demonstrates that the CSP is sufficient for an algebraic, Brouwer lattice to arise as L (G) for a suitable hyper-archimedean/-group G.

Before stating this theorem let us make the following useful observation: if L 1 and L 2 are two algebraic, Brouwer lattices and c(L1) is isomorphic to c (L2) , then L 1 is isomorphic to L 2.

3.2. THEOREM. Suppose L is an algebraic, Brouwer lattice with the CSP," there is a hyper-archimedean l-group G so that L(G)~-L.

Proof. By theorem 2.4 L has the F1P and the primes {p~ I 2cA} form a trivially ordered set. Let G* be the/-group of integer-valued functions on A with finite range; alternatively, the/-group of integral step functions on A. G* is hyper-archimedean, and so is any/-subgroup ([6], theorem 2.4). We define a mapping a:c ( L ) ~ G* by:

{~ if c~<pz; caz= if c~<p~.

Since two elements in c(L) are equal if and only if they have the same values a is a one to one map; moreover, e_< d in c (L) if and only if ca <_ da in G*. It is easy to verify that cr is a lattice embedding.

Now let G be the /-subgroup of G* generated by {co" [ c~c(L)}. In view of the remark preceding this theorem we will have done enough to show that P(G), the lattice of principal/-ideals of G, is isomorphic to c(L).

We define a new map z : e ( L ) ~ P (G) by letting cz = G (ca)= the principal/-ideal of G generated by ca. Once again it is easily verified that e n d if and only if cz~_dz, and so in particular z is one to one. In any/ -group/1 , H (a v b)=H (a)v H (b) and H (a A b )= H (a)c~ H (b) with 0 < a, b ~ H; so z is a lattice embedding as well.

The reader might make a note that the CSP has not been used yet in a direct way, only the FIP has. The presence of the CSP will make the map onto, and complete the proof, but without it we still seem to have some sort of an embedding. We shall return to this later.

A typical element of G has the form V~ A~ (~7=1 rn (c~, fl, i) (c(~, fl, i)) a), where the indicated joins and meets are taken over finite sets, and the rn~ are integers. If O<geG, and g is expressed as indicated, we may alter it to read: g = V, Ap [(~'=1 m(a, fl, i) (c(e, fl, i)) a ) v 0 ] , using the fact that the underlying lattice of G is distributive. What we must show is that G(g)=G(ca), for some cec(L); but it is sufficient to do this for each expression (~'=1 re(a, fl, i) (c(ct, fl, i ) ) a ) v 0. Without loss of generality we may therefore assume that g = (~7= 1 m,c~a)v 0. Using the CSP


we may replace ~ ' = 1 mici by a new linear combination in which the compact elements of L are pairwise disjoint. Assuming this has been done we then conclude that g = ~ ' = 1 m~c~, where each m i > 0. Now let c = c~ v c2 v . . . v c,; then G (g) = G (ca). This proves that �9 is onto P(G), and so c(L) ~- P(G); it follows that L ~- L(G), and we are done.

3.2.1. COROLLARY. I f G is a hyper-archimedean l-group, then one cannot tell from the lattice of l-ideals whether G is embeddable as an l-subgroup of an l-group of real valued functions with finite range.

Let us now return to the proof of theorem 3.2. I f L were just an algebraic, Brouwer lattice with the FIP we could still construct the embedding ~:c (L)- - )P (G). Now we must extend z to L:

3.3. LEMMA. Suppose L1 and Lz are algebraic lattices, and Lt satisfies the FIP; then if ~b:c(L,)--* L z is a lattice homomorphism, ~ has a unique extension to a map ~ : L, --+ L2 which preserves arbitrary joins. I f L 1 and L 2 a r e also Brouwer lattices, then this extension preserves finite intersections as well.

Proof. Define if:L1 -'->L2 by a ~ = V c~qS, where a = V c~, with c ~ c ( L ) . The only difficulty is to show that ~ is well-defined; after that the rest is easy. I f a = V c~= V dx, with cl, dx~c(L ), then each ci<_dzlvd~2v. . .vdx, for some 21 .. . . . 2,. Thus ci(~<<_dx,~ v ... v dx,~b; likewise each dz~ is below a finite join of the ci(a; it follows then that V cidp = V d~(a, and ~ is well defined.

Returning then to the embedding z:c ( L ) ~ P (G) and our remarks preceding lemma 3.3, �9 extends uniquely to a lattice embedding of L in L (G), preserving arbitrary joins. Since L (G) has the CSP we have an improvement on proposition 3.1.

3.1a. T H E O R E M . I f L is an algebraic, Brouwer lattice with the FIP then L can be embedded as a sublattice of an algebraic, Brouwer lattice with the CSP; the embedding is as a complete join sublattice.

The reader might well wonder about the presence of the FIP in the hypothesis of theorem 3.1 a: if q~:L, --+ L2 is an embedding of one algebraic lattice in another which preserves arbitrary joins, then if [c (L l)] ~b __ c(L2) and t 2 satisfies the FIP, so does L 1. For suppose c, d~c(L1) and c a d = V xi, with the x i upward directed; then cqS^dcb = V x ~ , and since c~b A d~b is compact in L z we have that cq~ A d~b = Xioq~, for some i o. But ~ is one to one, and so c A d = Xio , providing c A d is join-inaccessible and therefore compact.

In any case even this embedding is not as nice as it might seem. It may happen for instance that by embedding the hyper-archimedean, Brouwer lattice L as indicated above, one enlarges the set of meet-irreducibles; in the next section we show that this does actually happen. One would like then, an embedding of L in a hyper-archimedean, Brouwer lattice with the CSP which is 'dense' under some reasonable interpretation.


4. Examples and closing comments

First let us give two examples to show that the pair of conditions in b) of theorem 2.4 are independent.

Let X be an infinite set with the finite-complement topology, and let L = 0 (X), the lattice of open subjects of Z; L is an algebraic, Brouwer lattice if we interpret arbitrary meets as 'interior of set theoretic intersection'. Here L is hyper-archimedean, since the meet-irreducible open sets are simply the complements of singletons; further each element of L is compact, yet 0 (the empty set) is prime. Thus the primes do not form a trivially ordered set.

On the other hand let X = {xl, xz .... , x . . . . . , y, z}, and X ' = X \ { y , z}. Any subset of X' is open, and an open neighborhood o f y (resp. z) is one whose complement in X' is finite. Let L = 0 ( X ) ; once again L is a hyper-archimedean, Brouwer lattice. If U = X \ { y } and V = X \ { z } , then U and V are compact in L but U n V is not. However each prime is meet-irreducible. The author owes this example to Jed Keesling.

In the first example the Boolean algebra of polars is trivial, and every element s > 0 is basic. The second example also has a basis.

With regard to the remarks at the end of the previous section, let us apply the construction of theorem 3.2 to the lattice in the first example above. Suppose s and t are distinct meet-irreducibles of L, and let gt = (scr- ta) v 0. As in section 3, index the meet-irreducibles {t~ I ~.~A}; then

(gt)~={10 if t z = t otherwise.

Since t was arbitrary this shows that all of the finitely non-zero functions, (i.e. the cardinal sum of copies of integers indexed over A) are in G. This containment is proper since each tza has infinite support. The meet-irreducible /-ideals G~= = {g~G I g~=0} correspond to the tz in L in a one to one fashion. However, the /-ideal F of all finitely non-zero functions is proper, and hence is contained in some meet-irreducible/-ideal which is necessarily different from any of the Gz. So using the embedding z of section 3 one can 'create' new meet-irreducibles.

We close with some open questions: I. If L is a hyper-archimedean, Brouwer lattice without the CSP, does L still arise

as the lattice of o-ideals (directed, convex subgroups) of an abelian p.o. group? II. The hard question still is: what conditions does one have to place on an

archimedean, Brouwer lattice so that it is isomorphic to the lattice of/-ideals of some archimedean/-group? We can point to some necessary conditions: 1) the lattice must satisfy the FIP and 2) the set of primes is a root system. There is also the following necessary condition which appears to be stronger than 2): for each a, b e e ( L ) there

260 J. MARTINEZ

exist c, dec(L) with e ^ d=O, such that each value of a that exceeds b is a value of c, and each value of b that exceeds a is a value of d, and c<_a and d<_b.

In any case it seems hopeless to generalize the method of theorem 3.2. III. It should be interesting to consider whether the notion of an archimedean

lattice has any useful applications in algebra. For modules over a ring A one can investigate the effect of requiring lattices of submodules to be archimedean (resp. hyper-archimedean). If A is commutative, then every A-module has a hyper-archimedean lattice of submodules if and only if the ring is (von Neumann) regular. In the non-commutative case the rings can also be characterized. The author will present these results elsewhere.

IV. Finally, one can consider topologies on the sets of primes and the set of meet- irreducibles in an algebraic, Brouwer lattice L, and obtain dualities with certain cate- gories of topological spaces. Here again, the author will discuss these matters in a forthcoming article; in it the role played by hyper-archimedean lattices will be examined.

REFERENCES

l1 ] A. Bigard, Groupes archim~diens et hyper-archim~diens, No. 2 (1967-68). [2] G. Birkhoff, Lattice theory, Amer. Math. Soc. Coll. Publ., X X V (1967). [3] G. Birkhoff and O. Frink, Representations of lattices by sets, Trans. Amer. Math. Soc. 64 (1948),

299-316. [4] R. Bleier and P. Conrad, The lattice of closed ideals and a*-extensions o f an abelian l-group,

preprint. [5] P. M. Cohn, Universal algebra, Harper and Row (1965). [6] P. Conrad, Lattice ordered groups, Tulane University (1970). [7l P. Conrad, Epi-archimedean lattice ordered groups, preprint. [8] O. Frink, Pseudo-complements in semi-lattices, Duke Math. Jour. 29 (1962), 505-514. [9] G. Gr/itzer, Universal algebra, Van Nostrand (1968).

[10] G. Gr~itzer and E. T. Schmidt, Characterizations of congruence lattices of abstract algebras, Acta Sci. Math. 24 (1963), 34-59.

l11] J. P. Jans, Rings and homology, Holt, Rinehart and Winston (1964). [12] L. Nachbin, On a characterization of the lattice of all ideals o f a Boolean ring, Fund. Math..36

(1949), 137-142. [13] J. Varlet, Contribution ~ l~tude des treillis pseudo-complementds et des treillis de Stone, M6m.

Soc. Roy. Sci. Li6ge 8 (1963), 1-71.

University of Florida Gainesville, Florida U.S.A.

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