ALGEBRAIC AND CATEGORICAL PROPERTIES OF r-IDEAL SYSTEMS

The structures (G,r), where r is a system of ideals defined on a directed group G, play an important role in solving arithmetical problems. In this paper, we investigate how some properties of these systems are transferred in their cartesian products and their substructures. The results we obtain find an application in the study of categorical properties of these structures. 2000 Mathematics Subject Classification. Primary 06F15, 18A35; Secondary 20K25.


Introduction and preliminaries.
The theory of r -ideal systems defined on directed groups was firstly investigated by Lorenzen in 1939 (cf.[4]).Jaffard, in 1960 (cf.[1]), made a systematic study of these systems, which covers a large part of their properties, although the terminology he used was quite difficult, thus some of his results have been later rediscovered.These systems are important since a lot of arithmetical problems, such as the embedding of an integral domain into a greatest common divisor integral domain, the embedding of a po-group into a lattice-group, the investigation of Prüfer groups or Bezout domains, can be solved using their properties.
By an r -system of ideals in a directed po-group G we mean a map X X r (X r is called the r -ideal generated by X) from the set B(G) of all lower bounded subsets X of G into the power set of G, which satisfies the following conditions: (1) X ⊆ X r , (2) X ⊆ Y r ⇒ X r ⊆ Y r , (3) {a} r = a • G + = (a) for all a ∈ G, (4) a • X r = (a • X) r for all a ∈ G.An r -ideal is said to be finite if it is finitely generated, and said to be principal if it can be generated by one element.The set Ᏽ r (G) of the r -ideals of G, endowed with the multiplication is a commutative monoid, which contains the structure (Ᏽ f r (G), × r ), where Ᏽ f r (G) is the set of finite r -ideals, as a submonoid.In the following, a directed group G endowed with an r -system of ideals will be denoted by (G, r ).The structure (G, r ) has the following properties: (1) r -α total (respectively, finite) property if any (respectively, finite) r -ideal of G is principal.
(2) r -β total (respectively, finite) property if (4) r -δ total (respectively, finite) property if for any (respectively, finite) r -ideal X r of G, the transporter X r : We mention that (G, r ) has the r -δ total (respectively, finite) property if and only if for every x, y ∈ G and it follows that y ≤ x.Among all the r -systems defined on G, there exist two special ones, called the v-system and the t-system defined, respectively, by for any X ∈ B(G).
In the next section, we study how the above-mentioned properties of the structures (G 1 ,r 1 ) and (G 2 ,r 2 ) can be transferred into the cartesian product G 1 × G 2 and vice versa, considering that the directed group G 1 ×G 2 is endowed with a system of ideals denoted by r 1 ⊗ r 2 , (cf.[2]), where In addition, we make a similar research for the structures (G, r ) and (H, r ), where H is a directed subgroup of G and X r = X r ∩ H, for any X ∈ B(H).The system r will be mentioned as the restriction of r .Moreover, the results we derive find an application in the investigation of categorical properties.We recall some notions in order to specify the categorical approach we attempt.
A map f : -morphism if it is a group homomorphism and f (X r 1 ) ⊆ (f (X)) r 2 , for every X ∈ B(G 1 ).The map is a semigroup homomorphism and it will be mentioned as the map induced by f .We denote by K the category with objects (G, r ) and morphisms the (r 1 ,r 2 )-morphisms and by L the category with objects (Ᏽ r (G), × r ) and morphisms the semigroup homomorphisms.In [2], we have studied limits in the category K and we have proved that the map : for every object (G, r ) and every morphism f of K, is a functor which preserves the products.
In Section 3, we continue the study of the categories K, L and of the functor , in what concerns the existence of limits and the ability of to preserve or reflect them.Moreover, we define a proper subcategory L * of L, which is equivalent to K. We finish by defining subcategories of K and L according to the properties their objects have and we investigate limits in them as well as their relation via the above-mentioned functor.
2. Special structures with r -ideal systems.This section is devoted to the investigation of the properties of ideal systems.We denote by R(G) the set of all the rsystems defined on G and by R j (G) (respectively, R f j (G)), j = α, β, γ, δ, the subset of R(G) which contains the r -systems having the r −j total (respectively, finite) property, j = α, β, γ, δ.In the following, whenever we refer to a cartesian product G = G 1 × G 2 , we consider it endowed with the r 1 ⊗ r 2 -system, where r i ∈ R(G i ), i = 1, 2, and we denote by p i : G → G i , i = 1, 2, the usual projection maps.Especially, we prove that the properties a cartesian product G 1 ×G 2 possesses are determined by the properties its factors have and vice versa.
Proposition 2.1 (see [3]).Consider the structures (2.1) Proof.Since [3] has not yet been published, we mention that the isomorphism needed in the first congruence is defined by Proof.We distinguish the following cases: where thus, the structure (G 1 ,r 1 ) has the r 1 -α total property.In the same way, we prove that we consider the lower bounded subset ).In the same way, we prove that (G 2 ,r 2 ) has the r 2 -δ total property.
Proof.We observe that if X is a finite subset of G, then p i (X) is a finite subset of G i , i = 1, 2 and vice versa; if X i is a finite subset of G i , then we can always construct a finite subset X of G such that p i (X) = X i , for i = 1, 2. The result follows by arguing as in Proposition 2.2.
We can now prove proportionate results concerning subgroups of a directed group.Proposition 2.4.Consider the structures (G 1 ,r 1 ), (G 2 ,r 2 ), and f ,g : Proof.Obviously, the set E is a directed subgroup of G 1 , so the system r 1 is well defined.Let X be a lower bounded (respectively, finite) subset of E, that is, Proof.We denote by i : H → G the injection map, which is obviously an (r ,r )morphism and let i * : Ᏽ r (H) → Ᏽ r (G) be the induced semigroup homomorphism.If (G, r ) has the r -γ total property, then for every (2.7) Thus, the monoid Ᏽ r (H) is cancellative.Suppose now that (G, r ) has the r -δ total property and let X ∈ B(H) and a ∈ X r | X r .Then In the previous propositions, we have used the notion of a semigroup homomorphism induced by an (r 1 ,r 2 )-morphism.We shall prove that this kind of map does not include all semigroup homomorphisms from Ᏽ r 1 (G 1 ) to Ᏽ r 2 (G 2 ).Proposition 2.6.Consider the structures (G 1 ,r 1 ) and (G 2 ,r 2 ).There exist semigroup homomorphisms from Ᏽ r 1 (G 1 ) to Ᏽ r 2 (G 2 ), which are not induced by any (r 1 ,r 2 )- Proof.We prove this proposition by giving an example.Let (Z, +, ≤) be the additive group of the integers endowed with the usual ordering.Consider the cartesian product Z × Z, which becomes a partially ordered group with the componentwise ordering and addition.Put G 1 = (Z, +, ≤), G 2 = (Z × Z, +, ≤) and consider the structures (G 1 ,t 1 ), (G 2 ,t 2 ), where t 1 ,t 2 are the t-systems defined on G 1 ,G 2 , respectively.Then, from the definition of the t-system, it follows that where which means that the map f is well defined.Moreover, this map is a semigroup homomorphism, since for (2.11) Now suppose that the map f is induced by a (t 2 ,t 1 )-morphism f : It is obvious that the map f is a group homomorphism.In order to prove that it is not a (t 2 ,t 1 )-morphism, it would be enough to find a lower bounded subset and Hence, the map f is not a (t 2 ,t 1 )-morphism.

Categorical properties.
In [2], we have proved the existence of finite products and equalizers in the category K.More specifically, the product of the objects , where p 1 ,p 2 are the projection maps and the equalizer of the (r 1 ,r 2 )-morphisms f ,g : }, r 1 is the restriction of the r 1 -system into E and l : E → G 1 is the injection map.Proof.We can easily generalize the construction of finite products (cf.[2]) in order to verify that the product of an arbitrary family (G i ,r i ) i∈I of objects of K is the pair (( i∈I G i , r ), (p i ) i∈I ), where X r = i∈I (p i (X)) r i and p i , i ∈ I, the projection maps.Corollary 3.2.Consider the objects (G i ,r i ), i = 1, 2, 3, of K and the morphisms f : Proof.The proof is obvious, from the form the products and the equalizers have in the category K.
Proposition 3.3.The inclusion functor L → Sem does not reflect equalizers.
Proof.Consider the objects (G 1 ,t 1 ), (G 2 ,t 2 ) of the category K, as they have been defined in Proposition 2.6, the corresponding objects where p i , i = 1, 2, are the projection maps from Z × Z to Z. Let (E, l) be the equalizer of f and p * 1 in the category Sem, that is, If the inclusion functor L → Sem reflects equalizers, then there exists an object thus, p 1 (g) = p 1 (g) + p 2 (g).Hence, which is absurd.So, the equalizer (E, l) is not an object of the category L and this completes the proof.
The previous proposition shows that in the category L equalizers do not exist, in general.It is then natural for one to define a proper subcategory of L, which should have more properties.We put L * the subcategory of L, which has the same objects and as morphisms the semigroup homomorphisms induced by morphisms of the category K. Proof.It is obvious that this functor is well defined.In order to prove the equivalence of the categories K and L * , it is enough to observe that for every f ,g ∈ Hom K ((G 1 ,r 1 ), (G 2 ,r 2 )) with f = g, it follows that for any x ∈ G 1 .
Corollary 3.5.The category L * is complete and the functor : K → L * preserves and reflects limits.
In the following, we denote by K j (respectively, K f j ), the subcategory of K with objects (G, r ) which have the r −j total (respectively, finite) property and by L j (respectively, L f j ), the corresponding subcategories of L, for j = α, β, γ, δ.To avoid confusion, we symbolize the restriction of the functor : K → L, into the subcategories K j and K f j , by the same letter.We investigate the existence of limits in K j and K f j as well as the proportionate results for the categories L j and L f j .
(2) The categories K β and K f β have products.
Proof.Since all these categories are subcategories of K, it is enough to check whether the limits existing in K are reflected in them by the inclusion functor or not.The answer is obvious from Propositions 2.2, 2.3, 2.4, and 2.5.
It is obvious that the restriction of the functor : K → L into K j and K f j , j = α, β, γ, δ, preserves the products.We prove that the functor : K α → L α also preserves equalizers and pullbacks.Proposition 3.7.The functor : K α → L α preserves limits.

Proposition 3 . 1 .
The category K is complete.