A distributive semilattice not isomorphic to the maximal semilattice quotient of the positive cone of any dimension group

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Journal of Algebra 268 (2003) 290–300

www.elsevier.com/locate/jalgebr

A distributive semilattice not isomorphicto the maximal semilattice quotient

of the positive cone of any dimension group

Pavel Ružicka

Department of Algebra, Charles University, Sokolovská 83, 186 75 Prague 8, Czech Republic

Received 20 February 2002

Communicated by Kent R. Fuller

Abstract

We construct a distributive 0-semilattice which is not isomorphic to the maximal semilquotient of the positive cone of any dimension group. The size of the semilattice isℵ2. 2003 Elsevier Inc. All rights reserved.

Introduction

The notion of ideal appears in many algebraic theories. All ideals of a given algeobject form typically an algebraic lattice, usually modular, in many interesting cdistributive. It is natural to ask which algebraic lattices can be represented as ideal lof algebraic objects of a certain type.

It is well known that two-sided ideals of a von Neumann regular ring form an algedistributive lattice. Thus, one of the representation problems can be formulated as fowhich algebraic distributive lattices are isomorphic to the lattices of two-sided ideals oNeumann regular rings, in particular, as the lattices of two-sided ideals of locally maalgebras (i.e., direct limits of finite products of full matrix ringsMn(k) over a fieldk).

G.M. Bergman [1] constructed a locally matricial algebra with the ideal laisomorphic to an algebraic latticeL provided that either the latticeL is strongly distributiveor the set of its compact elements is at most countable. The author [7] proved tha

Research supported by grant MSN 113200007.E-mail address:[email protected].

0021-8693/$ – see front matter 2003 Elsevier Inc. All rights reserved.doi:10.1016/S0021-8693(03)00103-0

P. Ružicka / Journal of Algebra 268 (2003) 290–300 291

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algebraic distributive lattice whose semilattice of compact elements is closed undemeets is isomorphic to the lattice of two-sided ideals of some locally matricial algOn the negative side, F. Wehrung [10] constructed an algebraic distributive latticeis not isomorphic to the lattice of two-sided ideals of any von Neumann regularThe cardinality of the set of compact elements of his counterexample isℵ2. The sameauthor [11] proved that every algebraic distributive lattice with at mostℵ1 compactelements is isomorphic to the ideal lattice of some von Neumann regular ring. Howthe ring is not a locally matricial algebra (it is not even a unit regular ring) and thufollowing problem remains open.

Problem 1. Let L be an algebraic distributive lattice with at mostℵ1 compact elementsDoes there exist a locally matricial algebra with the lattice of two-sided ideals isomoto L? (Compare to [5, Problem 10.2] and [11, Problem 1].)

The lattice of two-sided ideals of a locally matricial algebraR is isomorphic to thelattice of ideals (i.e., directed convex subgroups) of its Grothendieck groupK0(R). Aslocally matricial algebras are direct limits of directed families of matricial algebrascorresponding Grothendieck groups are direct limits of directed families of simpgroups. The direct limits of directed families of simplicial groups were characterizeE.G. Effros, D.E. Handelman, and C.-L. Shen [2] to be unperforated, directed paordered Abelian groups satisfying the interpolation property. These groups aredimension groups.

K.R. Goodearl and D.E. Handelman [4] proved that any dimension group of smostℵ1 is isomorphic to the Grothendieck group of some locally matricial algebra wF. Wehrung [9] constructed a dimension group of sizeℵ2 which is not isomorphic to theGrothendieck group of any von Neumann regular ring.

In view of the result of K.R. Goodearl and D.E. Handelman, Problem 1 can be rein terms of dimension groups and their ideal lattices.

Problem 1′. Let L be an algebraic distributive lattice with at mostℵ1 compact elementsDoes there exist a dimension groupG with the lattice of ideals isomorphic toL? (See [5,Problem 10.1′].)

To prove that a given algebraic distributive latticeL occurs as the lattice of ideals ofdimension group, it suffices to show that the semilattice of compact elements ofL occurs asthe semilattice of finitely generated ideals of the group. The semilattice of finitely geneideals of a dimension group is isomorphic to the maximal semilattice quotient of its pocone.

In the paper, we construct a distributive 0-semilatticeT , of size ℵ2, which is notisomorphic to the maximal semilattice quotient of the positive cone of any dimegroup. We have not succeeded to modify the construction to get a counterexamsmaller cardinality, and so Problem 1′ remains open. Nevertheless, we answer negativmore general problem formulated in [5].

292 P. Ružicka / Journal of Algebra 268 (2003) 290–300

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Problem 10.1 [5] (Lifting distributive semilattices to dimension groups). Let S be adistributive 0-semilattice. Does there exist a dimension groupG such that the maximasemilattice quotient ofG+ is isomorphic toS?

1. Basic concepts

Lattices, semilattices

All semilattices considered in the text are join semilattices.Let L be a complete lattice and leta be an element ofL. Thena is calledcompact, if

for anyA⊆ L, a ∨

A implies thata ∨

F for some finiteF ⊆ A. The set of compacelements of a complete lattice is closed under finite joins, and thus forms a 0-semiNote that this set is not closed under finite meets in general.

A complete latticeL is calledalgebraic if every element ofL is a join of compactelements.

A nonempty lower subset of a semilatticeS closed under finite joins is called anidealof S. A latticeL is algebraic if and only if it is isomorphic to the lattice of all idealssome 0-semilattice [6, II.3. Theorem 13], in particular, if it is isomorphic to the latticall ideals of the semilattice of its compact elements.

Let Θ be a congruence relation of a semilatticeS. For x ∈ S, we denote by[x]Θ thecongruence class containingx. Given an idealI of a semilatticeS, we denote byΘ[I ] thesmallest congruence relationΘ of S under whichx ≡ y(Θ) for all x, y ∈ I . Note that foreveryu, v ∈ S, u≡ v(Θ[I ]) if and only if u∨ x = v ∨ x for somex ∈ I .

A semilatticeS is calleddistributive if for every a, b, c ∈ S such thatc a ∨ b thereexista′, b′ ∈ S with a′ a, b′ b, andc = a′ ∨ b′. A semilatticeS is distributive if andonly if its ideal lattice is distributive [6, II.5. Lemma 1].

Therefinement propertyis the following semigroup-theoretical axiom: A commutatsemigroupC satisfies therefinement propertyif and only if a0 + a1 = b0 + b1,(a0, a1, b0, b1 ∈ C) implies the existence ofcij ∈ C (i, j = 0,1) with ai = ci0 + ci1(i = 0,1) andbj = c0j + c1j (j = 0,1). A commutative monoid is arefinement monoidifit satisfies the refinement property. By induction we get immediately

Lemma 1.1. Let C be a commutative semigroup satisfying the refinement properx = x0

i + x1i (x, x0

i , x1i ∈G+) for all i < n, then there exists a setxf | f ∈ 2n of elements

of G+ such that

x0i =

∑f (i)=0

xf and x1i =

∑f (i)=1

xf

for everyi < n.

Note that semilattices are exactly idempotent commutative semigroups. On thehand, for a commutative semigroupC there exists a least congruence on C such thatC/ is a semilattice. The congruence is defined as follows: for everya, b ∈C, a b if


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ted

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and only if there exist a positive integern such thata nb andb na. The quotientC/is denoted by∇(C) and called themaximal semilattice quotient ofC.

It is an easy fact that a semilattice is distributive if and only if it satisfies the refineproperty. Consequently, if a commutative semigroupC satisfies the refinement properthen∇(C) is a distributive semilattice.

The correspondence that with a commutative monoidM associate∇(M) naturallyextends to a direct limits preserving functor from the category of commutative moto the category of semilattices (see [5, Section 2]). Forx ∈ M we denote by[x] thecorresponding element of∇(M).

A nonzero elementa of a semilatticeS is join-irreducibleif a = b∨ c impliesa = b ora = c for everyb, c ∈ S. We denote byJ(S) the partially ordered set of all join-irreducibelements of a semilatticeS. A distributive semilattice in which every element is a finite joof join-irreducible elements is calledstrongly distributive.The easy proof of the followinglemma is left to the reader.

Lemma 1.2. LetS be the0-semilattice of compact elements of an algebraic latticeL. Thenthe following conditions are equivalent:

(i) The semilatticeS is strongly distributive.(ii) The latticeL is isomorphic to the lattice of lower subsets of the partially ordered

J(S).(iii ) The latticeL is isomorphic to the lattice of lower subsets of some partially orde

set.

Dimension groups

The positive coneof a partially ordered Abelian groupG is a monoidG+ = a ∈G

| 0 a, and we putG++ = a ∈ G | 0 < a. If f :G→ H is an order preservinhomomorphism of partially ordered Abelian groupsG, H , thenf [G+] ⊆H+. We denoteby f+ the restriction off fromG+ to H+.

A partially ordered Abelian groupG is calledunperforatedif na 0 impliesa 0 forall a ∈G and every positive integern.

A partially ordered Abelian groupG is calleddirected if every element ofG is adifference of two elements ofG+. Note that a partially ordered Abelian group is direcif and only if it is directed as a partially ordered set.

A partially ordered setP satisfies theinterpolation propertyif for every a0, a1, b0,and b1 in P such thatai bj (i, j = 0,1), there existsc ∈ P such thatai c bj(i, j = 0,1). An interpolation groupis a partially ordered Abelian group which satisfithe interpolation property. A partially ordered Abelian group is an interpolation groand only if its positive cone is a refinement monoid [3, Proposition 2.1].

A dimension groupis an unperforated, directed, interpolation group. E.G. EffD.E. Handelman, and C.-L. Shen characterized dimension groups as direct limdirected diagrams of simplicial groups [2, Theorem 2.2], where asimplicial group is apartially ordered Abelian group isomorphic toZn with the positive cone(Z+)n for somepositive integern.


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A directed convex subgroup of a partially ordered Abelian group is called anideal.Wedenote byId(G) the lattice of all ideals of a partially ordered Abelian groupG, and byIdc(G) the semilattice of compact elements of the latticeId(G). It is easily seen that thsemilatticeIdc(G) is isomorphic to the maximal semilattice quotient∇(G+) of the positivecone of a partially ordered Abelian groupG.

An ordered vector space is a partially ordered Abelian group endowed with a strof vector space over the totally ordered fieldQ of rational numbers such that thmultiplication by a positive scalar is order-preserving. An ordered vector space whdirected and satisfies the interpolation property is called adimension vector space.Notethat every ordered vector space is unperforated. With a partially ordered Abelian grG

we associate the ordered vector spaceG ⊗Z Q with the positive coneG+ ⊗Z Q+. Thecorrespondence[a] → [a ⊗ 1] defines an isomorphism∇(G+)→ ∇(G+ ⊗Z Q+), andso every distributive 0-semilattice isomorphic to the maximal semilattice quotient opositive cone of some dimension group is, at the same time, isomorphic to the masemilattice quotient of some dimension vector space.

2. Weakly Archimedean partially ordered Abelian groups

Definition 2.1. A partially ordered Abelian groupG is calledweakly Archimedeanif foreverya, b ∈G+, there exists a positive integern such thatna b.

Definition 2.2. LetG be a partially ordered Abelian group, letV =G⊗Z Q be the orderedvector space associated toG. Fora, b ∈G+ we define

(a/b)= supq ∈Q+ | a⊗ 1 b⊗ q in V

.

Note that a partially ordered Abelian groupG is weakly Archimedean if and onlif for every a, b ∈ G++, (a/b) < ∞. We state some simple properties of the m(−/−) :G+ ×G+ → [0,∞].Lemma 2.3. LetG, H be partially ordered Abelian groups. Leta, a′, b, andb′ be elementsof G+. Then

(i) (a/b) > 0 if and only if[a] [b];(ii) if [a] = [a′] and[b] = [b′], then(a/b)=∞ if and only if(a′/b′)=∞;(iii ) if f :G→H is an order preserving homomorphism, then(f (a)/f (b)) (a/b).

Let G be a partially ordered Abelian group. Fora, b ∈ G+ we use the notationa⊥b ifc a, b implies thatc= 0 for everyc ∈G+.

Lemma 2.4. LetG be an interpolation group. Ifa⊥b (a, b ∈G+), thena, b c impliesthata + b c for everyc ∈G+.

Proof. SinceG is an interpolation group, there existsd ∈G such thata, b d a+ b, c.Then 0 a + b− d a, b, and sincea⊥b, we get thata + b= d c.


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Corollary 2.5. LetG be an interpolation group. Leta, b, c ∈G+ satisfya⊥b anda c.Then(c/b)= (c− a/b).

Lemma 2.6. Let c, d, b be elements of the positive cone of an ordered vector spaceV . If(c+ d/b)=∞ and(b/d) > 0, then(c/d)=∞.

Proof. From(b/d) > 0, it follows that there exists a positive integerm such thatd mb.Thus, for all positive integersn, m(n+1)b c+d m(c+b), whence(n+1)b c+b.Therefore,nb c. Lemma 2.7. LetP be a partially ordered upwards directed set. Let

〈G,fp :Gp→G〉p∈P = lim−−→〈Gp,fp,q :Gp→Gq〉pq in P

in the category of partially ordered Abelian groups. Letp ∈ P . Then, for everya, b ∈G+p

(i) if p q in P , then(a/b) (fp,q(a)/fp,q(b));(ii) (fp(a)/fp(b))= sup(fp,q(a)/fp,q(b)) | p q in P .

Moreover, ifP is σ -directed(i.e., every countable subset ofP is bounded), then thereexistsq in P abovep, such that

(fp(a)/fp(b)

)= (fp,q(a)/fp,q(b)

).

In particular, if (fp(a)/fp(b))=∞, then there existsq in P abovep, with (fp,q (a)/

fp,q(b))=∞.

3. Example

In this section, we construct in two steps a distributive 0-semilattice which isisomorphic to the maximal semilattice quotient of the positive cone of any dimegroup. First we define a strongly distributive 0-semilatticeS which is not isomorphic to themaximal semilattice quotient of the positive cone of any weakly Archimedean dimegroup. The cardinality of the semilatticeS is ℵ2. Then, by means of the semilatticeS, weconstruct a distributive 0-semilatticeT , of the same cardinalityℵ2, which is not isomorphicto the maximal semilattice quotient of the positive cone of any dimension group.

In connection with the first step of our construction, note that from the resuG.M. Bergman [1], it follows that every strongly distributive 0-semilattice is isomorpto the maximal semilattice quotient of the positive cone of some dimension gK.R. Goodearl and F. Wehrung defined for every distributive latticeD a dimension vectospaceQ〈D〉 such that∇(Q〈D〉+) is isomorphic toD as a join semilattice [5, Section 4It is readily seen that the dimension vector spaceQ〈D〉 is weakly Archimedean. Furthethey proved that every countable distributive 0-semilattice is isomorphic to the masemilattice quotient of the positive cone of some countable dimension vector


space

alension

nsion

ouphat

,

[5, Theorem 5.2]. We begin with proving that also in this case the dimension vectorcan be taken weakly Archimedean.

Theorem 3.1. Every countable distributive0-semilattice is isomorphic to the maximsemilattice quotient of the positive cone of some countable weakly Archimedean dimvector space.

Proof. Let S be a countable distributive 0-semilattice. Letσ :S → ∇(Q+) be the〈∨,0〉-homomorphism defined byσ(0) = [0] while σ(x) = [1] for x > 0. By [8,Theorem 7.1], there are a countable dimension vector spaceG, a positive homomorphismf :G→ Q, and an isomorphismε :S → ∇(G+) such thatσ = ∇(f+) ε. For everya, b ∈G++, 0< f (a), f (b), and, by Lemma 2.3(iii),(a/b) (f (a)/f (b))= f (a)/f (b).It follows thatG is weakly Archimedean. Theorem 3.2. There exists a strongly distributive semilatticeS, of sizeℵ2, which is notisomorphic to the maximal semilattice quotient of any weakly Archimedean dimegroup.

Proof. We defineS to be a strongly distributive 0-semilattice with

J(S)= z ∪ yβ | β < ω1 ∪ xα | α < ω2

ordered by the relationsz > yβ > xα (α < ω2, β < ω1).To the contrary, suppose that there exists a weakly Archimedean dimension grG

and an isomorphismφ :∇(G+)→ S. Without loss of generality, we can assume tG is a dimension vector space. Pickc, bβ (β < ω1), andaα (α < ω2) from G+ suchthat (φ[c])= z, φ([bβ]) = yβ for everyβ < ω1, andφ([aα]) = xα for everyα < ω2. Inaddition, we can choosebβ so thatc > bβ , thus,(c/bβ) 1 for everyβ < ω1.

For all α < ω2, β < ω1, put µα,β = (bβ/aα)/3. Note that 0< µα,β <∞ for everyα < ω2, β < ω1. Let Ω be the family of all nonempty finite subsets ofω1. Choose a mapF :ω2→Ω so that

∀α < ω2:∑

β∈F(α)

µα,β > (c/aα). (3.1)

Observe thatcard(F−1(B))=ℵ2 for someB = β0, . . . , βn−1 ∈Ω . For eachi < n, putb1i = bβi andb0

i = c− bβi . Sincec > bβi , b0i ∈G+ for everyi < n, and so, by Lemma 1.1

there existbf (f ∈ 2n) in G+ such that

b0i =

∑f (i)=0

bf and b1i =

∑f (i)=1

bf (i < n).

Letχi (i < n) denote the characteristic function ofi. Further, denote byD the set of mapsf :n→ 2 that satisfyf (i)= 1= f (j) for at least two different non-negative integersi, j


smaller thann. If f ∈D, then[bf ] lies below at least two elements of the set[bβi ] | i < n,hence there exists a finite subsetAf of ω2 such that

[bf ] =∨α∈Af

[aα].

PutA=⋃f∈D Af and pickγ ∈ F−1(B)\A. Sinceaγ ⊥aα for everyα ∈A,

aγ ⊥∑f∈D

bf ,

thus,a fortiori, for everyi < n,

aγ ⊥∑

f∈D,f (i)=1

bf .

By Corollary 2.5, for everyi < n,

3µγ,βi =(b1i /aγ

)=(b1i −

∑f∈D,f (i)=1

bf /aγ

)= (bχi /aγ ).

Therefore

c=∑

f : n→2

bf ∑i<n

bχi >∑i<n

2µγ,βiaγ ,

and, by (3.1),

∑i<n

2µγ,βiaγ = 2

(∑i<n

µγ,βi

)aγ > 2(c/aγ )aγ .

Consequentlyc > 2(c/aγ )aγ , hence(c/aγ ) 2(c/aγ ), which is not the case.Let L be the sublattice ofP(ω1) consisting of all subsetsx of ω1 where

x =⋃i<n

[αi,βi)

for some natural numbern and 0 α0 < β0 < · · ·< αn−1 < βn−1 ω1. Put

B = x ∈ L | ∃α ∈ ω1: x ⊆ [0, α),

U = y ∈ L | ∃α ∈ ω1: y ⊇ [α,ω1)

.

Note thatB is an ideal of the latticeL andU is the dual filter of the idealB.


alp. Let

tgroup.

l

Let S be a 0-semilattice. PutS∗ = S\0, and set

T = (0 ×B)∪ (

S∗ ×U)⊆ S ×L. (3.2)

Lemma 3.3. If S is a distributive0-semilattice, then the semilatticeT defined by(3.2) isdistributive.

Proof. Let x y ∨ z for somex = (x1, x2), y = (y1, y2), andz= (z1, z2) ∈ T . We provethat there existy ′ y, z′ z ∈ T with x = y ′ ∨ z′.

In casex1= 0, we puty ′2= x2∧ y2, z′2= x2∧ z2. Sincex2 ∈B andB is an ideal ofL,y ′2, z′2 ∈ B. Theny ′ = (0, y ′2), z′ = (0, z′2) ∈ T andx = y ′ ∨ z′.

Let x1 > 0. Since the semilatticeS is distributive, there existy ′1 y1, z′1 z1 in S withx1= y ′1∨z′1. First assume that one of the elementsy ′1, z′1, sayz′1, equals 0. Theny ′1 > 0, andsoy2 ∈ U , hence there existsα < ω1 such thaty2⊇ [α,ω1), whencex2 y2∨(z2∩[0, α)).We puty ′2= x2∧ y2 andz′2= x2 ∧ (z2 ∩ [0, α)). Theny ′2 ∈ U , z′2 ∈ B, andy ′ = (x ′1, y ′2),z′ = (0, z′2) are elements ofT with x = y ′ ∨ z′. If 0 < y ′1, z′1, thenx2, y2, z2 ∈ U , hencey ′2 = x2 ∧ y2, z′2 = x2 ∧ z2 ∈ U . By the definition, bothy ′ = (y ′1, y ′2), z′ = (z′1, z′2) areelements ofT andx = y ′ ∨ z′. Lemma 3.4. Let S be a distributive0-semilattice which is not isomorphic to the maximsemilattice quotient of the positive cone of any weakly Archimedean dimension grouT be the semilattice defined by(3.2). ThenT is a distributive0-semilattice which is noisomorphic to the maximal semilattice quotient of the positive cone of any dimensionMoreovercard(T )= card(S).

Proof. By Lemma 3.1,card(S) ℵ1, and so the equalitycard(T )= card(S) is clear. ThesemilatticeT is distributive by Lemma 3.3.

Forα < ω1 put

Bα =x ∈L | x ⊆ [0, α).

Givenα < ω1, setIα = 0 ×Bα , and denote byI the ideal0 × B of the semilatticeT .For eachα < ω1 put Tα = T/Θ[Iα], and letϕα,β (α < β < ω1) stand for the canonicaprojectionTα → Tβ . Finally, denote byϕα (α < ω1) the canonical projectionsTα →T/Θ[I ]. Since I is a union of the increasing chainIα | α < ω1 of ideals of thesemilatticeT , we see that

⟨T/Θ[I ], ϕα

⟩α<ω1

= lim−−→〈Tα,ϕα,β〉α<β<ω1.

Note that for every(s, x), (t, y) ∈ T , (s, x) ≡ (t, y)(Θ[Iα]) if and only if s = t andx ∩ [α,ω1) = y ∩ [α,ω1), and (s, x) = (t, y)(Θ[I ]) if and only if s = t . Hence thecorrespondence(s, x)Θ[I ] → s defines an isomorphism fromT/Θ[I ] ontoS.

To the contrary, suppose there exists a dimension groupG and an isomorphismι :∇(G+)→ T . Put


s

ts

nt

eof the

Jα =a − b | a, b ∈G+ andι

([a]), ι([b])∈ Iα

(α < ω1),

J = a − b | a, b ∈G+ andι

([a]), ι([b])∈ I.

For eachα < ω1 denote byGα the quotientG/Jα . Let fα,β (α < β < ω1) stand for thecanonical projectionGα → Gβ , and denote byfα (a < ω1) the canonical projectionGα → G/J . The idealJ is the union of the increasing chainJα | α < ω1 of ideals ofthe groupG, hence

〈G/J,fα〉α<ω1 = lim−−→〈Gα,fα,β〉α<β<ω1.

For everyα < ω1 denote byια the isomorphism∇(G+α ) ontoTα sending[a + Jα] toι([a])Θ[Iα]. It is easy to see that the diagram

∇(G+α

)∇(

f+α,β)

ιαTα

ϕα,β

∇(G+β

) ιβTβ

commutes for everyα < β < ω1. Since the functor∇ preserves direct limits, there exisan isomorphismλ :∇((G/J )+)→ S $ T/Θ[I ] such that for everya ∈ G+ and every(s, x) ∈ T , ι([a])= (s, x) implies thatλ([a + J ])= s.

Since, due to Lemma 3.2,S is not isomorphic to the maximal semilattice quotieof the positive cone of any weakly Archimedean dimension group, there exista′, b′ ∈(G/J )++ such that(a′/b′) = ∞. Put s = λ([a′]), t = λ([b′]), and picka, b ∈ G+ sothat ι([a]) = (s, [0,ω1)), and ι([b]) = (t, [0,ω1)). Then [a + J ] = [a′], [b + J ] = [b′],hence, by Lemma 2.3(ii),(a + J/b + J )=∞. By Lemma 2.7, there existsα < ω1 with(a + Jα/b+ Jα)=∞. Note that

ια([a + Jα]

)= (s, [0,ω1)

)Θ[Iα], ια

([b+ Jα])= (

t, [0,ω1))Θ[Iα].

Pick c, d ∈ G++ with ι([c]) = (s, [α + 1,ω1)), andι([d]) = (0, [0, α + 1)). Then[a] =[c + d], hence(c + d + Jα/b + Jα) =∞ by Lemma 2.3(ii). Further,[b] > [d], whichimplies (b + Jα/d + Jα) > 0. Applying Lemma 2.6, we get that(c + Jα/b + Jα) =∞,whence

(s, [α + 1,ω1)

)Θ[Iα] = ια

([c+ Jα]) ια

([b+ Jα])= (

t, [0,ω1))Θ[Iα]. (3.3)

Observe that for every(s, x), (t, y) ∈ T , (s, x)Θ[Iα] (t, y)Θ[Iα] if and only if s t

andx ∩ [α,ω1)⊇ y ∩ [α,ω1). Thus, according to (3.3),[α + 1,ω1)⊇ [α,ω1), which is acontradiction. Theorem 3.5. There exists a distributive0-semilatticeT which is not isomorphic to thmaximal semilattice quotient of the positive cone of any dimension group. The sizesemilattice isℵ2.


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of the

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(1999)

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Proof. By Theorem 3.2 there exists a distributive 0-semilatticeS, of sizeℵ2, which isnot isomorphic to the maximal semilattice quotient of the positive cone of any weArchimedean dimension group. Apply Lemma 3.4.

Acknowledgments

The author wishes to thank Friedrich Wehrung for his interest and careful readingmanuscript.

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A distributive semilattice not isomorphic to the maximal semilattice quotient of the positive cone of any dimension group