ORDERED GROUPS WITH GREATEST COMMON DIVISORS THEORY

An embedding (called a GCD theory) of partly ordered abelian group G into abelian l-group Γ is investigated such that any element of Γ is an infimum of a subset (possible non-finite) from G. It is proved that a GCD theory need not be unique. A complete GCD theory is introduced and it is proved that G admits a complete GCD theory if and only if it admits a GCD theory G Γ such that Γ is an Archimedean l-group. Finally, it is proved that a complete GCD theory is unique (up to o-isomorphisms) and that a po-group admits the complete GCD theory if and only if any v-ideal is v-invertible.


Introduction.
The problem of embedding of partly ordered abelian directed group (po-group, shortly) G into an abelian lattice ordered group (l-group) Γ is connected with investigation of arithmetical properties of rings, monoids and po-groups and it has its motivation in the theory of algebraic numbers.Various embedding theorems 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 Šafarevič [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-group G is said to admits some "divisor theory" if there exists an l-group Γ and a homomorphism h : G / / Γ such that (1) h is an ordered embedding (i.e., o-monomorphism), (2) h(G) is dense in Γ (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 Šafarevič.
(∀α ∈ Γ ) ∃g 1 ,...,g n ∈ G, α = h(g 1 ) ∧•••∧h(g n ), (1.1) (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, if Γ is required to be a free abelian group Z (P ) for some set P then the conditions (1.1), (1.2), and (1.3) are equivalent and h is then called the theory of divisors of G.For general l-group Γ the condition (1.1) characterizes po-groups with the so-called theory 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 the strong 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-group G with the embedding h : G / / Γ which satisfies the condition (1.3).He proved also that the group of divisibility of an integral domain R satisfies the condition (1.3) if and only if R is v-domain on the contrary to the result of Geroldinger and Močkoř [7] stating that the group of divisibility of R satisfies the condition (1.1) if and only if R is a Prüfer v-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 uniqueness of the GCD theory.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.
It is clear that if G admits a quasi divisor theory it admits the GCD theory, as well.In fact, let h : G / / Γ be the quasi divisor theory of G and let for some a i ,b j ∈ G, we have a i ∈ B and b j ∈ A for all i, j and it follows that α = β.
The principal tool for investigation of properties of po-groups with GCD theory seems to be the notion of an r -ideal.Recall that by an r -system of ideals in a po-group G we mean a map X X r (X r is called an r -ideal) from the set of all lower bounded subsets X of G into the power set of G which satisfies the following conditions: (1) and it is called a t-system, if (2.2) An r -system r is said to be of a finite character, if X r = Y ⊆X,Y finite Y r .An r -ideal X r is finitely generated if X r = Y r for some finite subset Y .Clearly, any t-system is of finite character.On the set Ᏽ r (G) of r -ideals we may define an ordering by X r ≤ Y r if and only if Y r ⊆ X r and a multiplication For any r -closed po-group G we may construct another r -system in G, denoted by r a , such that r a is regularly closed, i.e., in the semigroup (Ᏽ f r (G), ×) of all finitely generated r -ideals the cancellation law holds.The quotient group Λ r (G) For t and v systems we have t a = t, v a = v and the Lorenzen r -group for r = t or r = v then consists of all formal quotients A r /B r , where A and B are finite sets.A homomorphism f : (G, r ) / / (H, s) of po-groups with ideal systems r and s, respectively, is called The theory of r -ideals of po-groups seems to be a tool which enables us to establish relationships between arithmetical properties of integral domains and the theory of po-groups.There exists a lot of works justifying these processes, one of the last result in this direction being a paper of Geroldinger and the author (see [7]) where it is proved that properties of being a Prüfer v-multiplication domain, a domain of Krull type, or an independent domain of Krull type, are purely multiplicative ones and can be expressed by using r -ideals in the corresponding groups of divisibility.
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.Let h : For any subset X in a po-group G we write g ≤ X for g ≤ x for all x ∈ X and by ∧X we understand ∧ x∈X x.Proposition 2.2.Let G be a po-group and let h : G / / Γ be an o-monomorphism into an l-group Γ .Then the following statements are equivalent. ( For the proof of this proposition we need the following lemma.
Proof.Let A ⊆ G be lower bounded and let g ∈ A t .Then there exists a finite subset

Then we have h(y) ≤ h(K) ≤ h(K)
and it follows that y ≤ K. Since g ∈ K t , we have g ≥ y and it follows that y −1 ∈ h(g) −1 .Then according to the condition (2), h(g) ≥ h(K) and h(g) ∈ (h(K)) v .Therefore, h is a (t, t)-morphism.
(5)⇒(6).Trivial.( 6)⇒(3).Let A be a lower bounded subset in G and let α = h(A).Then A ⊆ ᾱ.We show that α = h( ᾱ).Let β ≤ h( ᾱ), then β ≤ h(A) and we have β ≤ α.Hence, α = h( ᾱ). ( 6)⇒ (7).Let α ∈ Γ and let A ⊆ G be a lower bounded subset such that α = h(A).We prove that α = h(A t ).It is clear that for any lower bound β of h(A t ), β is a lower bound of h(A) and it follows that β ≤ α.Hence, we have to prove only that α ≤ h(A t ).Let g ∈ A t .Then there exists a finite subset K ⊆ A such that g ∈ K t and for The following proposition has its origin in the paper of Clifford [4] who originaly proved that for a theory of divisors h : G / / Z (P ) the set ᾱ is v-ideal for any α ∈ 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.

Proposition 2.4. Let h : G
/ / Γ be a GCD theory.Then for any α ∈ Γ and any lower bounded subset A ⊆ G such that α = h(A) we have ᾱ = A v .
Therefore, ᾱ = ᾱv .Now let α = h(A).Then A ⊆ ᾱ and A v ⊆ ᾱv = ᾱ.Conversely, for g ∈ ᾱ and for any y ≤ A we have h(y) ≤ α ≤ h(g) and it follows that y ≤ g.Hence, The following proposition is a generalization of the above-mentioned result of Clifford.

Proposition 2.5. Let h : G
/ / Γ be a GCD theory.Then the map¯: (Γ , •) / / (Ᏽ v (G), ×) such that α ᾱ is an o-isomorphism into and the following diagram commutes: On the other hand, we obtain Then according to Proposition 2.4, we have (2.6) Therefore, α • β = ᾱ × β and ¯is a group homomorphism.Moreover, from Proposition 2.2 it follows that it is injective and o-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].
Lucius pointed out that integral domain with GCD theory need not be Prüfer vmultiplication domain.For this he used the example of v-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.
Lemma 2.7.Let R be an integral domain and let G(R) be its group of divisibility.Then there exists an isomorphism σ : Proof.Let K be the quotient field of R and let w : [15]), we have w(a+b) ∈ (w(a), w(b)) v ⊆ A v and it follows that σ (A) is an ideal.It is clear that σ is a homomorphism.Conversely, for J ∈ Ᏽ v (R) we set ψ(J) = w(J × ).Then ψ(J) is v-ideal and ψ is a homomorphism which is inverse to σ .
Let R be a v-domain which is not Prüfer v-multiplication domain.Then according to previous lemma every finitely generated v-ideal of G = G(R) is v-invertible, but finitely generated v-ideal of G do not form a group.Hence, G does not admit quasi divisor theory according to [2] or [7].On the other hand, G admits GCD theory as the following proposition shows (a version of this proposition for integral domains was proved by Lucius [10]).Proposition 2.8.Let G be a po-group and let any finitely generated v-ideal of G be v-invertible.Then the embedding h : (G) in this case and elements of Λ v (G) can be identified with some v-ideals of G.In fact, for any formal fraction In quasi divisor theory it is well known that the embedding G / / Λ t (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].
Then if G admits a GCD theory the same is true for G .
Proof.Let X ⊆ G be a finite set and let A ⊆ G be a finite set such that σ (A) = X. (2.8) and G admits a GCD according to Proposition 2.8.
For any embedding h : G / / Γ of a po-group G into an l-group we can construct the divisor class group Ꮿ h = Γ /h(G) of this embedding.Hence, we have the following short exact sequence (2.9) For po-groups with quasi divisor theory this divisor class group has special importance, since the density condition of h can be substitute by some density condition of ϕ h (see [12,16]).In the following proposition, we show that even for po-groups with GCD this short exact sequence has some density property.
(2) For any lower bounded subset A ⊆ G there exists h(A) in Γ .

Proof. (2)⇒(1). Let A v ∈ Ᏽ v (G) and let α = h(A).
Then A ⊆ ᾱ and since ᾱ is v-ideal in G, we have A v ⊆ ᾱ.Let g ∈ ᾱ and let b be a lower bound of A. Then h(g) ≥ α ≥ h(b) and we have g ≥ b.Therefore, g ∈ A v and A v = ᾱ.
A GCD theory of a po-group G that satisfies the equivalent conditions of Proposition 3.1 will be called a complete GCD theory.It should be observed that there are examples of GCD theory h : G / / Γ such that { ᾱ : α ∈ Γ } ⊂ Ᏽ v (G).In fact, for any non complete l-group Γ the identity map Γ / / Γ is such example, as follows from the previous proposition.

Although for a GCD theory
can be a proper inclusion, the set of v-ideals of the form ᾱ is even in this case dense in Ᏽ v (G).In the following proposition, we show that even for general GCD theory, Ᏽ v (G) represents a lattice completion of the set { ᾱ : α ∈ Γ }.Moreover, from Proposition 2.5 it follows that (Γ , •, ≤) is isomorphic to ({ ᾱ : α ∈ Γ }, ×, ≤) and from the next proposition it follows that Ᏽ v (G) represents the lattice completion of the lattice (Γ , ∧).Proposition 3.5.Let h : G / / Γ be a GCD theory.Then the following statements hold.
(2) For any v-ideal A v in G there exists a lower bounded subset Φ in Γ such that A v = α∈Φ ᾱ.
Proof.(1) Let β be a lower bound of Φ. From Proposition 2.2 it follows that there exists b ∈ G such that h(b) ≤ β and it follows that b is a lower bound of α∈Φ ᾱ.Then it is clear that ( α∈Φ ᾱ) v = α∈Φ ᾱ. ( Then it is clear that Φ = ∅ since, for example, for any finite K ⊆ A we have h(K) ∈ Φ.Then α∈Φ ᾱ ⊆ A v is lower bounded and we have α∈Φ ᾱ Proof.From Proposition 2.5 it follows that¯is an o-isomorphism into.Let A v ∈ Ᏽ v (G).Then from Proposition 3.5 and corollary of Proposition 3.1 it follows that A v = α∈Φ ᾱ = α∈Φ α.Hence,¯is surjective and complete l-homomorphism.Further, we show that for any A v ∈ Ᏽ v (G), h(A v ) = h(A).In fact, let α = h(A), then according to Proposition 2.4, we have ᾱ = A v .Hence, for any a ∈ A v we have h(a) ≥ α and it follows that α = h(A) ≥ h(A v ) ≥ α.Hence, χ(A v ) = h(A v ) = h(A).Moreover, χ is a group homomorphism.In fact, we have Corollary 3.7.Let G be a po-group.Then G admits a complete GCD theory if and only if any v-ideal of G is v-invertible.
Proof.Let G admits a complete GCD theory.Then according to the previous Proposition 3.6, (Ᏽ v (G), ×) is a group.Conversely, let Ᏽ v (G) be a group.Then it is clear that it is a complete l-group, where for any lower bounded subset Φ ⊆ Ᏽ v (G), the set B = Av ∈Φ A is lower bounded in G and B v = Av ∈Φ A v .Moreover, for any v-ideal A v we have A v = ( a∈A (a) v ) v = a∈A (a) v .Therefore, the embedding (•) v : G / / Ᏽ v (G) is a complete GCD theory according to Propositions 2.2 and 3.1.
Corollary 3.8.Any po-group G admits at most (up to o-isomorphisms) one complete GCD theory.
Proof.Let h : G / / Γ be a complete GCD theory.Since Γ is a complete l-group, it is Archimedean and according to Proposition 3.6, Γ is o-isomorphic to Ᏽ v (G).

3 .Proposition 3 . 1 .
Groups with complete GCD theories.For a GCD theoryh : G / / Γ the set Ᏽ v (G) of all v-ideals of G has a special subset, namely { ᾱ : α ∈ Γ } ⊆ Ᏽ v (G),as follows from Proposition 2.4.A natural question is when this subset equals to Ᏽ v (G).This problem is solved by the following proposition.Let h : G / / Γ be a GCD theory.Then the following statements are equivalent.

. Basic properties of groups with GCD theory. The
This uniqueness property is closely connected with the Lorenzen t-group Λ notion of GCD theory was introduced by Lucius for integral domains only.We present firstly the translation of his definition for po-groups.
t (G) of G, since for any quasi divisor theory h : G / / Γ the natural embedding G / / Λ t (G) is the theory of quasi divisors, as well, and Λ t (G) is o-isomorphic to Γ .For GCD theory h : G / / Γ the situation is a little more complicated.Although the embedding G / / Λ t (G) is a GCD theory (in case that G admits the GCD

Definition 2.1. Let
G be a po-group.By a GCD theory of G we mean a homomorphism h of G into an abelian l-group Γ such that (1) h is an o-monomorphism, (