© Hindawi Publishing Corp.
ORDERED GROUPS WITH GREATEST
COMMON DIVISORS THEORY
JIˇRÍ MOˇCKOˇR
(Received 15 October 1999)
Abstract.An embedding (called aGCD theory) of partly ordered abelian groupGinto abelian l-group Γ is investigated such that any element ofΓ is an infimum of a subset (possible non-finite) fromG. It is proved that a GCD theory need not be unique. Acomplete GCD theoryis introduced and it is proved thatGadmits a complete GCD theory if and only if it admits a GCD theoryG
//
Γ such thatΓ is an Archimedean l-group. Finally, it is proved that a complete GCD theory is unique (up too-isomorphisms) and that a po-group admits the complete GCD theory if and only if anyv-ideal isv-invertible.Keywords and phrases. po-groups, l-groups, GCD-theory, complete GCD-theory.
2000 Mathematics Subject Classification. Primary 13F05, 06F15.
1. Introduction. The problem of embedding of partly ordered abelian directed group (po-group, shortly)Ginto an abelian lattice ordered group (l-group)Γ is con-nected with investigation of arithmetical properties of rings, monoids and po-groups and it has its motivation in the theory of algebraic numbers. Various embedding the-orems not only for po-groups, but also for integral domains (e.g., embedding into the GCD domain) might serve as a basis for various concepts in arithmetical investigation of rings, po-groups and monoids.
The axiomatic concept of a domain with this embedding property was formulated firstly in the book of Borewicz and Šafareviˇc [3]. After then this axiomatic concept was firstly transformed onto semigroups with cancellation by Skula [16, 17] and then transformed and generalized onto po-groups and monoids by several authors (see [2, 6, 7, 8, 13, 14]). Nevertheless in all these transformations and generalizations the principal structure of this axiomatic concept remains the same and for po-groups it could be expressed as follows.
A po-groupGis said to admits some “divisor theory” if there exists an l-groupΓand a homomorphismh:G
//
Γ such that(1) his an ordered embedding (i.e.,o-monomorphism),
(2) h(G)isdenseinΓ (in some specific sense).
For example, during the history of development of this problem the following principal formulations of the density condition (2) appeared, most of them being due to Skula, Arnold and Borewicz and Šafareviˇc.
(∀α∈Γ) ∃g1,...,gn∈G, α=h(g1)∧···∧h(gn), (1.1)
(∀α,β∈Γ) {g∈G:h(g)≥α} = {g∈G:h(g)≥β} ⇒ α=β, (1.3) (∀α,β∈Γ) (∃γ∈Γ)such thatα∧γ=1, β·γ∈h(G). (1.4)
In po-groups and integral domains theory these conditions (1.1), (1.2), (1.3), and (1.4) characterize po-groups with various types of divisor theory or integral domains which are generalizations of Krull domains (see [2, 7, 13, 16]). For example, ifΓ is required to be a free abelian groupZ(P)for some setPthen the conditions (1.1), (1.2), and (1.3) are equivalent andhis then called thetheory of divisors ofG. For general l-groupΓ
the condition (1.1) characterizes po-groups with the so-calledtheory of quasi divisors
(this notion was introduced by Aubert [2], although Jaffard investigated properties of such groups more early [9]), and the condition (1.4) leads to po-groups with thestrong
theory of quasi divisors(see [14]). (This last axiom was originaly formulated by Arnold
[1] and later used by Skula [16].)
Although the conditions (1.2) and (1.3) were not investigated for general l-group separately, Aubert [2] mistakes (1.1) and (1.3) (and, hence (1.4)) for equivalent. This mistake was pointed out by Lucius [10] who proved (in language of integral domains) that (1.3) does not imply (1.1), in general, and he introduced (again in the language of integral domains) a new notion of a po-group with the GCD theory as a po-groupG with the embeddingh:G
//
Γ which satisfies the condition (1.3). He proved also that the group of divisibility of an integral domainRsatisfies the condition (1.3) if and only ifRisv-domain on the contrary to the result of Geroldinger and Moˇckoˇr [7] stating that the group of divisibility ofRsatisfies the condition (1.1) if and only ifRis a Prüferv-multiplication domain. Hence, po-groups which satisfy the condition (1.3) really represent a generalization of po-groups with quasi divisor theory.In this paper, we want to describe some properties of po-groups with GCD theory. The principal problem we want to solve concerns the uniquenessof the GCD the-ory. It is well known (and for the first time it was proved by Jaffard [9]) that up to o-isomorphism for any po-group there exist as most one quasi divisor theory. This uniqueness property is closely connected with the Lorenzent-groupΛt(G)ofG, since for any quasi divisor theory h:G
//
Γ the natural embeddingG//
Λt(G) is the theory of quasi divisors, as well, andΛt(G)is o-isomorphic toΓ. For GCD the-oryh:G//
Γ the situation is a little more complicated. Although the embedding G//
Λt(G)is a GCD theory (in case thatGadmits the GCD theory), as well, it is the open problem stated by Lucius [10] whetherΛt(G)is again (up too-isomorphism) unique within l-groupsΓ with the GCD theoryh:G//
Γ. In this paper, we show that, in general, the GCD theory is not unique which answers the Lucius question in negative. We further introduce the notion of acomplete GCD theoryand we prove that Gadmits the complete GCD theory if and only if it admits a GCD theoryh:G//
Γsuch thatΓ is Archimedean. Moreover, we show that a po-group admits the complete GCD theory if and only if anyv-ideal isv-invertible and we prove that a complete GCD theory is at most unique.
Definition2.1. LetGbe a po-group. By a GCD theory ofGwe mean a homomor-phismhofGinto an abelian l-groupΓ such that
(1) his ano-monomorphism,
(2) (∀α,β∈Γ) α=β{g∈G:h(g)≥α} = {g∈G:h(g)≥β}.
It is clear that ifGadmits a quasi divisor theory it admits the GCD theory, as well. In fact, leth:G
//
Γ be the quasi divisor theory ofGand letA= {g∈G:h(g)≥α} = {g∈G:h(g)≥β} =Bforα,β∈Γ. Sinceα=h(a1)∧ ··· ∧h(an),β=h(b1)∧
··· ∧h(bm)for someai,bj∈G, we haveai∈Bandbj∈Afor alli,j and it follows thatα=β.
The principal tool for investigation of properties of po-groups with GCD theory seems to be the notion of anr-ideal. Recall that by anr-system of ideals in a po-group Gwe mean a mapXXr (Xr is called anr-ideal) from the set of all lower bounded subsetsXofGinto the power set ofGwhich satisfies the following conditions:
(1) X⊆Xr,
(2) X⊆Yr ⇒ Xr⊆Yr,
(3) {a}r=a·G+=(a)r∀a∈G, (4) a·Xr=(a·X)r∀a∈G.
Recall that anr-system is called av-system, if Xv=
X⊆(y)r,y∈G
(y)r, (2.1)
and it is called at-system, if
Xt=
Y⊆X,Yfinite
Yv. (2.2)
Anr-systemris said to be ofa finite character, ifXr=Y⊆X,YfiniteYr. Anr-idealXr
isfinitely generatedifXr =Yr for some finite subsetY. Clearly, anyt-system is of
finite character. On the setᏵr(G)ofr-ideals we may define an ordering byXr≤Yrif and only ifYr⊆Xr and a multiplicationXr×Yr=(XrYr)r=(X·Y )r. Anr-idealAris calledr-invertibleif it has the inverse in the semigroupᏵr(G). A po-groupGwith an
r-systemr isr-closed if(Xr:Xr)⊆G+ for anyXr. For anyr-closed po-groupGwe may construct anotherr-system inG, denoted byra, such thatraisregularly closed, i.e., in the semigroup(Ᏽfr(G),×)of all finitely generatedr-ideals the cancellation law holds. The quotient groupΛr(G) ofᏵfr(G)is called theLorenzenr-group ofGand a maph:G
//
Λr(G)defined byh(g)=(g)r is ano-isomorphism into. Fortandvsystems we haveta=t,va=vand the Lorenzenr-group forr=torr=vthen consists of all formal quotientsAr/Br, whereAandBare finite sets. A homomorphism
f:(G,r )
//
(H,s)of po-groups with ideal systemsrands, respectively, is called (r ,s)-morphism if for any lower bounded subsetX⊆Gwe havef (Xr)⊆(f (X))s.We will now investigate some simple properties of po-groups with GCD theory. The density property from Definition 2.1 is equivalent to some other conditions as the following proposition states, where we used the following notation. Leth:G
//
Γbe ano-monomorphism. Forα∈Γ we put ¯αh=α¯= {g∈G:h(g)≥α} =h−1((α) v∩
h(G)). For any subsetXin a po-groupGwe writeg≤X forg≤xfor allx∈Xand by∧Xwe understand∧x∈Xx.
Proposition2.2. LetGbe a po-group and leth:G
//
Γbe ano-monomorphisminto an l-groupΓ. Then the following statements are equivalent.
(1) (∀α,β∈Γ)α¯=β¯α=β,
(2) (∀α,β∈Γ)α¯⊆β¯β≤α,
(3) (∀α∈Γ) α=h(α)¯ ,
(4) (∀α∈Γ) (α)v=(h(G)∩(α)v)v, (5) (∀α∈Γ) (∃Av∈Ᏽv(G)) α=h(Av),
(6) (∀α∈Γ) (∃A⊆G) Ais lower bounded andα=h(A),
(7) (∀α∈Γ) (∃At∈Ᏽt(G)) α=h(At).
For the proof of this proposition we need the following lemma.
Lemma2.3. Leth:G
//
Γ be ano-monomorphism into an l-groupΓ such that itsatisfies the condition (2) from Proposition 2.2. Thenhis a(t,t)-morphism.
Proof. LetA⊆Gbe lower bounded and letg∈At. Then there exists a finite subset
K⊆Asuch thatg∈Kt=Kv and we haveh(g)∈(h(K))v =(h(K))v. In fact, we show that(h(K))−1⊆h(g)−1. Lety−1∈(h(K))−1. Then we haveh(y)≤h(K)≤
h(K)and it follows that y ≤K. Sinceg∈Kt, we have g ≥y and it follows that
y−1∈h(g)−1. Then according to the condition (2),h(g)≥h(K)andh(g)∈(h(K))v. Therefore,his a(t,t)-morphism.
Proof of Proposition2.2. (1)⇒(2). Let ¯α⊆β¯. We show that in this caseα∨β=
¯
α. Clearly,α∨β⊆α¯. Forg∈α¯⊆β¯it follows thath(g)≥α∨βand ¯α⊆α∨β. From (1) then followsα≥β.
(2)⇒(3). Letα∈Γ. We show firstly that(α)v∩h(G)= ∅. In fact, let us consider the following only possible cases.
(a)α≥1Γ. Assume that the statement is not true. Then (α2)v∩h(G)= ∅and it follows that ¯α=α2. Therefore, we haveα=α2according to (2) and it follows that
α=1Γ. But in this case we haveh(1G)∈(1Γ)v∩h(G), a contradiction. (b)α <1Γ. Then we haveh(1G)∈(1Γ)v∩h(G).
(c)α1Γ. SinceΓ is directed, there existsβ∈Γ+ such thatα·β≥1Γ and it follows
thath(g)∈(α·β)v∩h(G)for someg∈G. Henceh(g)≥α·β≥α.
Now letβ∈Γ be such thatβ≤h(α)¯ . It follows that ¯α⊆β¯and according to (2) we haveα≥β. Therefore,α=(h(α))¯ .
(3)⇒(4). Letα∈Γ. Sinceh(α)¯ is lower bounded byα, there exists thev-ideal(h(α))¯ v. We have to prove only thatα∈(h(α))¯ v. Letβ≤h(α)¯ . From (3) it follows thatβ≤α and we haveα∈(h(α))¯ v.
(4)⇒(5). Letα∈Γ. Then ¯αis lower bounded inG. In fact, since the statement (2) holds, analogously as we did in the proof of implication (2)⇒(3) we can prove that h(g−1)∈(α−1)v∩h(G)for someg∈G. Hence,h(g)≤αand it follows thatg≤α¯ inG. Since (4)(3), we haveα=h(α)¯ . We prove thatα=h(α¯v). In fact, since ¯
α⊆α¯v, then any lower bound of ¯αv is less than or equal toα. It remains to show thatα≤h(α¯v). Letg∈α¯v and leth(y)≤α, hence, ¯α⊆(y)v. Sinceg∈α¯v, theng is contained in any principalv-ideal containing ¯α, hence, we haveg≥y. Therefore, h(y)≤α∧h(g)and it follows thatα−1⊆(α∧h(g))−1. Since the opposite inclusion holds and (4)(1), we haveα≤h(g)andα≤h(α¯v).
(5)⇒(6). Trivial.
(6)⇒(3). LetA be a lower bounded subset inG and let α=h(A). Then A⊆α¯. We show thatα=h(α)¯ . Letβ≤h(α)¯ , thenβ≤h(A)and we haveβ≤α. Hence, α=h(α)¯ .
(6)⇒(7). Letα∈Γ and letA⊆Gbe a lower bounded subset such thatα=h(A). We prove thatα=h(At). It is clear that for any lower boundβofh(At),β is a lower bound of h(A)and it follows that β≤α. Hence, we have to prove only that α≤h(At). Letg∈At. Then there exists a finite subsetK⊆Asuch thatg∈Kt and forβ=h(K)∈Γ we haveh(g)∈h(Kt)⊆(h(K))t=(β)t, according to the lemma. Hence,h(g)≥β≥α=h(A).
(7)⇒(3). Letα=h(At)for some lower bounded subsetA⊆G. ThenAt⊆α¯and it follows thatα=h(At)≥h(α)¯ ≥α. Therefore,α=h(α)¯ .
The following proposition has its origin in the paper of Clifford [4] who originaly proved that for a theory of divisorsh:G
//
Z(P)the set ¯αis v-ideal for anyα∈ Z(P). In language of integral domains this proposition was proved for GCD theory by Lucius [10]. Due to the importance of this result for our next results we translate this statement into the language of po-groups and present a little different proof, although the proof is, in fact, contained in the proof of Proposition 2.2.Proposition2.4. Leth:G
//
Γ be a GCD theory. Then for anyα∈Γ and anylower bounded subsetA⊆Gsuch thatα=h(A)we haveα¯=Av.
Proof. We show firstly that ¯αis anv-ideal. From the proof of Proposition 2.2 it follows that ¯αis lower bounded inG. Letg∈α¯v. Then according to Proposition 2.2,
g∈α¯v=
y∈G ¯
α⊆(y)v=h(y)
(y)v=
y∈G α≥h(y)
(y)v. (2.3)
Thenα−1=(h(g)∧α)−1. In fact, lety ∈α−1. Then α≥h(y−1)and we haveg∈
(y−1)
v. Thus, h(y−1)≤α∧h(g) and it follows that y ∈(α∧h(g))−1. The other inclusion is trivial. Then from Proposition 2.2 then it follows thatα≤h(g)andg∈α¯. Therefore, ¯α=α¯v. Now letα=h(A). ThenA⊆α¯andAv⊆α¯v=α¯. Conversely, for
g∈α¯and for anyy≤Awe haveh(y)≤α≤h(g)and it follows thaty≤g. Hence, g∈Av andAv=α¯.
Proposition2.5. Leth:G
//
Γ be a GCD theory. Then the map¯:(Γ,·)//
(Ᏽv(G),×)such thatαα¯is ano-isomorphism into and the following diagram
com-mutes:
Γ −
//
Ᏽv(G)G
h
OO
G.
(·)v
OO
(2.4)Proof. Letα,β∈Γ. Then according to Proposition 2.2, we haveα·β=h(α·β). On the other hand, we obtain
α·β=h(α)¯ ·h(β)¯=h(α¯·β).¯ (2.5)
Then according to Proposition 2.4, we have ¯
α×β¯=α¯v×β¯v=
hα¯·β¯, α·β=α·β v=
hα·β . (2.6)
Therefore,α·β=α¯×β¯and ¯ is a group homomorphism. Moreover, from Proposition 2.2 it follows that it is injective ando-isomorphism into. The commutativity of the diagram follows from(g)v=h(g).
The following corollary (in integral domains language) is due to Lucius [10]. Corollary2.6. Leth:G
//
Γ be a GCD theory. Then any finitely generatedv-ideal ofGisv-invertible.
Proof. Let g1,...,gn ∈ G and let α =h(g1)∧ ··· ∧h(gn). Then according to Proposition 2.4, we have ¯α=(g1,...,gn)v and it follows that
1Ᏽv(G)=1G v=α·α−1=α¯×α−1. (2.7)
Hence, ¯αisv-invertible.
Lucius pointed out that integral domain with GCD theory need not be Prüferv -multiplication domain. For this he used the example ofv-domain which is not Prüfer v-multiplication domain constructed in [11]. The following lemma enables us to use this example to show that po-groups with GCD theory need not admit quasi divisor theory.
Lemma2.7. Let R be an integral domain and let G(R) be its group of
divisibil-ity. Then there exists an isomorphism σ : (Ᏽv(G(R)),×)
//
(Ᏽv(R),×) such thatσ (Ᏽfv(G(R)))=Ᏽfv(R).
Proof. LetKbe the quotient field ofR and letw:K×
//
G(R)be the corre-sponding semi-valuation ofR. ForA∈Ᏽv(G(R))we set σ (A)=w−1(A)∪ {0K}. Leta,b∈σ (A). Sincew(a+b)≥αfor allα∈G(R)such thatα≤w(a),w(b)(see [15]), we havew(a+b)∈(w(a),w(b))v⊆Avand it follows thatσ (A)is an ideal. It is clear thatσ is a homomorphism. Conversely, forJ∈Ᏽv(R)we setψ(J)=w(J×). Then
LetRbe av-domain which is not Prüferv-multiplication domain. Then according to previous lemma every finitely generatedv-ideal of G=G(R)isv-invertible, but finitely generatedv-ideal ofGdo not form a group. Hence,Gdoes not admit quasi divisor theory according to [2] or [7]. On the other hand,Gadmits GCD theory as the following proposition shows (a version of this proposition for integral domains was proved by Lucius [10]).
Proposition2.8. LetGbe a po-group and let any finitely generatedv-ideal ofG
bev-invertible. Then the embeddingh:G
//
Λv(G)is the GCD theory ofG.Proof. It should be observed that the Lorenzen groupΛv(G)exists, sinceGisv -closed. Moreover,Λv(G)=Λt(G)in this case and elements ofΛv(G)can be identified with somev-ideals ofG. In fact, for any formal fractionAv/Bv ∈Λv(G) (withA,B finite) there exists v-ideal Cv such thatAv =Bv×Cv. In this case, Av/Bv can be identified withCv. Hence, letCv,Dvbe such thatCv=Dv, whereXv= {g∈G:(g)v⊆
Xv}. But in this caseCv=Dvand(·)v:G
//
Λv(G)is the GCD theory.In quasi divisor theory it is well known that the embeddingG
//
Λt(G)is (up to isomorphisms) the only possible quasi divisor theory (if it exists). In the next section, we show that for GCD theory it is not true. This answers in negative the question presented by Lucius [10].Proposition2.9. Letσ:G
//
Gbe a surjective(v,v)-morphism of po-groups.Then ifGadmits a GCD theory the same is true forG.
Proof. LetX⊆Gbe a finite set and letA⊆Gbe a finite set such thatσ (A)=X. LetCv be thev-inverse ofAvinG,(1G)v=(A·C)v. Then we have
1G∈σ1G v =σ(A·C)v ⊆(σ (A·C))v=Xv×(σ (C))v. (2.8)
Thus,(1G)v ⊆Xv×(σ (C))v. SinceA·C ⊆(1G)v, we haveX·σ (C)⊆σ ((1G)v)⊆
(1G)v and it follows thatXv×(σ (C))v ⊆(1G)v. Therefore,Xv isv-invertible inG andGadmits a GCD according to Proposition 2.8.
For any embeddingh:G
//
Γ of a po-groupGinto an l-group we can constructthedivisor class groupᏯh=Γ/h(G)of this embedding. Hence, we have the following
short exact sequence
1
//
G h//
Γ ϕh//
Ꮿh//
0. (2.9)For po-groups with quasi divisor theory this divisor class group has special impor-tance, since the density condition ofhcan be substitute by some density condition of ϕh(see [12, 16]). In the following proposition, we show that even for po-groups with GCD this short exact sequence has some density property.
Proposition2.10. Leth:G
//
Γ be a GCD theory. Then for anyα∈Γ,α >1, we haveProof. We show firstly that for anyβ∈Γ there existsγ∈Γ+ such thatϕh(β)=
ϕh(γ). In fact, according to the proof of Proposition 2.2,(β−1)v∩h(G)= ∅andγ=
β·h(g)≥1 for someg∈G. Henceϕh(β)=ϕh(γ).
Letϕh(β)∈Ꮿh. Then from the above statement if follows that−ϕh(β)∈ϕh(Γ+) and there existsδ∈Γ+such that−ϕh(β)=ϕh(δ). Moreover, there existsγ∈Γ\(α)v such thatδ·γ∈h(G). In fact, let us consider the following only possible cases.
(a)αδ orα > δ. Then from Proposition 2.2 it follows that h(δ)¯ \h(α)¯ = ∅. Let h(g)∈h(δ)¯ \h(α)¯ . Thenh(g)∈(α)v and there existsγ∈Γ+such thath(g)=δ·γ andγ∈Γ+\(α)v.
(b)α≤δ. Thenα·δ > δ≥α and h(α·δ)⊂h(δ)¯. In fact, if the inclusion is not proper, we haveα·δ=h(α·δ)=h(δ)¯ =δaccording to Proposition 2.2, a contra-diction. Leth(g)∈h(δ)¯ \h(α·δ). Thenh(g)=γ·δfor someγ≥1. Ifγ≥α, then we haveh(g)≥α·δ, a contradiction. Hence,γ∈Γ+\(α)v.
From this property we obtainϕh(δ)+ϕh(γ)=0 andγ∈Γ+\(α)v. Hence,ϕh(Γ+)⊆
ϕh(Γ+\(α)v).
3. Groups with complete GCD theories. For a GCD theoryh:G
//
Γ the set Ᏽv(G)of allv-ideals ofGhas a special subset, namely{α¯:α∈Γ} ⊆Ᏽv(G), as follows from Proposition 2.4. A natural question is when this subset equals toᏵv(G). This problem is solved by the following proposition.Proposition3.1. Leth:G
//
Γbe a GCD theory. Then the following statements are equivalent.(1)For anyv-idealAv inGthere existsα∈Γ such thatAv=α¯.
(2)For any lower bounded subsetA⊆Gthere existsh(A)inΓ.
(3)Γ is a complete l-group.
Proof. (2)⇒(1). LetAv ∈Ᏽv(G) and letα=h(A). Then A⊆α¯ and since ¯αis
v-ideal in G, we haveAv ⊆ α¯. Letg∈ α¯ and let b be a lower bound ofA. Then
h(g)≥α≥h(b)and we haveg≥b. Therefore,g∈Av andAv=α¯.
(1)⇒(3). LetΦ⊆Γ be a lower bounded subset and letβ≤Φ for some β∈Γ. For any α∈Φ we have α=h(α)¯ according to Proposition 2.2. Thenα∈Φα¯ is lower
bounded inG. In fact, according to Proposition 2.2,β−1=h(β−1)≤h(r−1)for some
r−1∈β−1. Then for anyg∈α∈Φα¯ we haveh(g)≥α≥β≥h(r )andr is a lower bound ofα∈Φα¯. LetAv=(α∈Φα)¯ v. Then according to (1), there existsγ∈Γ such that ¯γ=Av. We haveγ=ΦinΓ. In fact, for anyα∈Φwe have ¯α⊆ρ∈Φρ¯⊆Av=γ¯ and from Proposition 2.2 it follows thatα≥γ. Further, letω≤αfor allα∈Φ. Then ¯
α⊆ω¯ andα∈Φα¯⊆ω¯. Sinceωisv-ideal, we have ¯γ=Av ⊆ω¯ and it follows that
γ≥ωby Proposition 2.2. (3)⇒(2). Trivial.
Corollary3.2. Leth:G
//
Γ be a complete GCD theory. Then h is a (v,v) -morphism.Proof. LetA⊆Gbe a lower bounded set. Then forα=h(A)we haveh(Av)=
h(α)¯ ⊆(h(A))v=(α)v=(h(A))vas follows from Proposition 3.1.
Corollary3.3. Leth:G
//
Γ be a complete GCD theory. Then for any lowerbounded subsetΦ⊆Γ we have
α∈Φ α=
α∈Φ
¯
α. (3.1)
Proof. For any β≤Φ in Γ there exists b∈G such that h(b)≤β. Thenb is a lower bound ofα∈Φα¯and according to Proposition 3.1, there existsγ∈Γ such that
¯
γ=(α∈Φα)¯ v=α∈Φα¯. We prove thatγ=α∈Φα. Since ¯α⊆γ¯, it follows thatγ≤α
for anyα∈Φ. For anyω∈Γ such thatω≤Φwe haveα∈Φα¯⊆ω¯ and since ¯ωis a v-ideal, we have ¯γ⊆ω¯. Hence,γ≥ω.
Theorem3.4. LetGbe a po-group. ThenGadmits a complete GCD theory if and
only if it admits a GCD theoryh:G
//
Γ such thatΓ is an Archimedean l-group.Proof. Lethbe a GCD theory such thatΓ is Archimedean. It is well known (cf. [5]) that for any abelian Archimedean l-groupΓ its completionΓcan be constructed such that
(1) Γ is a complete l-group, (2) Γ is an l-subgroup ofΓ,
(3) Γ is dense inΓ, i.e., for any 1<a∈Γthere existsα∈Γ such that 1< α≤a, (4) For anya∈Γ we havea={β∈Γ:β≥a}.
We observe further that if∆⊆Γis such thatα=∆exists inΓ, thenαis the infimum of∆inΓ, as well, i.e., we need not to distinguish infimum operations inΓ andΓ. In fact, letα=∆inΓ and leta∈Γbe such thata≤∆. Then we haveα∨a≤∆and we have to show thatα∨a=α. Suppose, by the way of contradiction thatα∨a> α. Then we have 1< (α∨a)·α−1. SinceΓis dense inΓ, there existsβ∈Γ such that 1< β≤(α∨a)·α−1. Hence, for anyω∈∆we haveω≥α∨a≥β·α > α, a contradiction.
Now, leth:G h
//
Γ//
Γ be the composition ofhand the embedding ofΓ intoΓ. We show thathis a complete GCD theory. Leta∈Γ. Thena=β∈Φβ, whereΦ=
{β∈Γ :β≥a}. Sincehis a GCD theory, for anyβ∈Φwe haveβ=h(β)¯ =h(β)¯ and it follows that ¯a=β∈Φβ¯. In fact, forg∈¯awe haveh(g)≥aandh(g)∈Φ,g∈ h(g)⊆β∈Φβ¯(the other inclusion is trivial). Then we havea=h(¯a)=h(β∈Φβ)¯.
In fact, letb∈Γ be such thatb≤h(β∈Φβ)¯. Thenb≤h(β)¯ for anyβ∈Φand we have b≤h(β)¯ =h(β)¯ =β. Therefore,b≤β∈Φβ=aandhis a GCD theory according
to Proposition 2.2.
Conversely, ifGadmits a complete GCD theoryh:G
//
Γ, thenΓ is a complete l-group and it follows thatΓ is Archimedean.Although for a GCD theoryh:G
//
Γ the inclusion relation{α¯:α∈Γ} ⊆Ᏽv(G) can be a proper inclusion, the set ofv-ideals of the form ¯αis even in this casedensein Ᏽv(G). In the following proposition, we show that even for general GCD theory,Ᏽv(G) represents a lattice completion of the set{α¯:α∈Γ}. Moreover, from Proposition 2.5 it follows that(Γ,·,≤)is isomorphic to({α¯:α∈Γ},×,≤)and from the next proposition it follows thatᏵv(G)represents the lattice completion of the lattice(Γ,∧).Proposition3.5. Leth:G
//
Γbe a GCD theory. Then the following statements hold.(1)For any lower bounded subsetφ⊆Γ there existsα∈Φα¯inᏵv(G).
(2)For any v-idealAv inG there exists a lower bounded subset Φ in Γ such that
Av=α∈Φα¯.
Proof. (1) Letβbe a lower bound ofΦ. From Proposition 2.2 it follows that there existsb∈Gsuch thath(b)≤βand it follows thatbis a lower bound ofα∈Φα¯. Then
it is clear that(α∈Φα)¯ v=α∈Φα¯.
(2) LetAv∈Ᏽv(G). We setΦ= {α∈Γ:Av⊆α¯}. Then it is clear thatΦ= ∅since, for example, for any finiteK⊆Awe haveh(K)∈Φ. Thenα∈Φα¯⊆Avis lower bounded and we haveα∈Φα¯=(α∈Φα)¯ v=Av. In fact, the inclusion⊆is trivial. Leta∈Av, then for anyg∈h(a)we haveg∈(a)v⊆Avand it follows thath(a)∈Φ. Therefore,
a∈h(a)⊆(α∈Φα)¯ v.
Proposition3.6. Leth:G
//
Γbe a complete GCD theory. Then¯:(Γ,·,≤)//
(Ᏽv(G),×,≤)is a complete l-isomorphism with the inverse l-isomorphismχsuch that
χ(Av)=h(Av)=h(A)for anyAv∈Ᏽv(G).
Proof. From Proposition 2.5 it follows that¯is ano-isomorphism into. LetAv∈ Ᏽv(G). Then from Proposition 3.5 and corollary of Proposition 3.1 it follows that
Av=α∈Φα¯=α∈Φα. Hence,¯is surjective and complete l-homomorphism. Further,
we show that for anyAv∈Ᏽv(G),h(Av)=h(A). In fact, letα=h(A), then ac-cording to Proposition 2.4, we have ¯α=Av. Hence, for anya∈Av we haveh(a)≥α and it follows that α=h(A)≥h(Av)≥α. Hence, χ(Av)=h(Av)=h(A). Moreover,χis a group homomorphism. In fact, we have
χAv×Bv =χ(A·B)v =
h(A·B)=h(A)·h(B)=χAv ·χBv . (3.2)
Further,χand¯are mutually inverse maps. In fact, forα∈Γ we haveχ(α)¯ =h(α)¯ =
αas follows from Proposition 2.2, and forAv∈Ᏽv(G)we haveχ(Av)=h(Av)=
h(A)=Av, according to Proposition 2.4. Finally,χis a complete l-homomorphism. In fact, letᏵ⊆Ᏽv(G)be a lower bounded subset and letΦ=χ(Ᏽ)⊆Γ. ThenΦis lower bounded and forα=χ(Av)we have ¯α=Av. Then we obtain
χ
Av∈Ᏽ Av
=χ
α∈Φ
¯ α =χ
α∈Φ α
=
α∈Φ α=
Av∈Ᏽ
χAv (3.3)
Corollary3.7. LetGbe a po-group. ThenGadmits a complete GCD theory if and
only if anyv-ideal ofGisv-invertible.
Proof. Let G admits a complete GCD theory. Then according to the previous Proposition 3.6,(Ᏽv(G),×)is a group. Conversely, letᏵv(G)be a group. Then it is clear that it is a complete l-group, where for any lower bounded subsetΦ⊆Ᏽv(G), the set B=Av∈ΦA is lower bounded inG and Bv =
Av∈ΦAv. Moreover, for any v-idealAv we haveAv =(a∈A(a)v)v=a∈A(a)v. Therefore, the embedding(·)v:
G
//
Ᏽv(G)is a complete GCD theory according to Propositions 2.2 and 3.1. Corollary3.8. Any po-groupGadmits at most (up too-isomorphisms) one com-plete GCD theory.Proof. Leth:G
//
Γ be a complete GCD theory. SinceΓ is a complete l-group, it is Archimedean and according to Proposition 3.6,Γ iso-isomorphic toᏵv(G).acknowledgment. The research was partly supported by the GA ˇCR (201/97/ 0433).
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Jiˇrí Moˇckoˇr: Department of Mathematics, University of Ostrava, CZ-702 00, Ostrava, Czech Republic