#### Abstract

Let be a bounded Hilbert algebra and a -closed subset of . The Hilbert algebra of fractions is studied regarding maximal and irreducible deductive systems. As important results, we can mention a necessary and sufficient condition for a Hilbert algebra of fractions to be local and the characterization of this kind of algebras as inductive limits of some particular directed systems.

#### 1. Introduction

The positive implication algebras characterize positive implicative logic, as it is mentioned in . The theory of this kind of algebras has been developed by Diego in . He also called them Hilbert algebras.

The notion of deductive system, defined by Monteiro, is equivalent to the notion of implicative filter used by Rasiowa in . The maximal and irreducible deductive systems play an important role in the study of these algebras as the representation theorem for Hilbert algebras (see Theorem 4.4) states.

Maximal deductive systems are studied by Buşneag in . He is also the one who studied the Hilbert algebras of fractions with respect to a -closed subset in [4, 5].

This paper is structured as follows. In “Preliminaries” are presented some fundamental results concerning Hilbert algebras and deductive systems. Next, there are established some properties referring to maximal and irreducible deductive systems. These particular deductive systems are connected to -closed subsets. It is also defined the local Hilbert algebra and we provide a necessary and sufficient condition for a Hilbert algebra to be local. The following section presents the spaces and . In the last section we study the Hilbert algebras of fractions, specially the one corresponding to maximal deductive systems. We prove that many results from ring theory maintain for Hilbert algebras as well. For example, we characterize a Hilbert algebra of fractions to be local. Finally, using as a guide-line the construction of the dual of the category of bounded distributive lattice, presented in , we define a presheaf on the base of the topological space and we prove that a local Hilbert algebra of fractions is isomorphic to an inductive limit of a directed system. For irreducible deductive systems we offer a similar result.

#### 2. Preliminaries

Definition 2.1. Let be a non empty set, a binary operation on and
A triplet is called Hilbert algebra if the following axioms hold, for each :
(),(),() implies .

A Hilbert algebra becomes a poset by defining an order relation such that if and only if . is the largest element of with respect to this order.

If the algebra has a smallest element, denoted by , it is called a bounded Hilbert algebra.

Definition 2.2. If are bounded Hilbert algebras, is said to be a morphism of Hilbert algebras if and

Theorem 2.3 (see ). In a Hilbert algebra the following relations hold for each :(1), (2),(3),(4),(5),(6), (7), (8),(9)if , then and We define and where .

Theorem 2.4 (see ). In a bounded Hilbert algebra , the following relations hold for each of the elements :(1) and ,(2),(3) and ,(4),(5) implies that ,(6),(7); , (8); ; ; ,(9),(10),(11),(12), .(13)

Definition 2.5. A subset of a Hilbert algebra is called a deductive system if and from it results that .

In a Hilbert algebra for Jun defines in  the deductive system Using this notion, he gives the following characterization of deductive systems.

Theorem 2.6 (see ). Let be a nonempty subset of a Hilbert algebra. Then, is a deductive system of if and only if for any

We denote the set of all deductive systems in with .

If is a subset of , the deductive system generated by is the least deductive system containing . We will denote it by .

Theorem 2.7 (see ). Let be a Hilbert algebra and a subset of . The deductive system generated by is the set of the elements for which there exist such that

In particular, if we consider the set the deductive system is called the deductive system generated by .

In  it is proved that is a Heyting algebra with respect to the following operations:

,,

Corollary 2.8 (see ). If is a Hilbert algebra, , and , then

In  is defined the concept of ideals in a Hilbert algebra as follows.

Definition 2.9 (see ). A nonempty subset of a Hilbert algebra is called an ideal of if:(1), (2) for all , (3) for all and

In [11, 12] it is proved that the notions of deductive systems and ideals in a Hilbert algebra are equivalent. Also, the final result from  states that every congruence on a Hilbert algebra is uniquely determined by its kernel which is a deductive system. Hence, ideals, deductive systems and congruence kernels in a Hilbert algebra coincide.

In  it is proved much more: is an algebraic lattice which is distributive and isomorphic with the lattice of all congruences on

Definition 2.10 (see ). Let be a Hilbert algebra and a proper deductive system. is called irreducible if for any two proper deductive systems such that either or

Definition 2.11 (see ). A deductive system of a Hilbert algebra is said to be maximal if it is proper and it is not a proper subset of any proper deductive system in .

For a Hilbert algebra we will denote the set of all maximal deductive systems of by and the set of all irreducible deductive systems of with According to their definitions, it is obvious that

Theorem 2.12 (see ). Let be a Hilbert algebra.(i)If is a proper deductive system and there exists an irreducible deductive system such that and (ii)If , there exists an irreducible deductive system such that and (iii)If is bounded, each proper deductive system is included in a maximal one.

Lemma 2.13 (see ). Let be a bounded Hilbert algebra. The following statements are equivalent:(i) is a maximal deductive system in (ii)For and , either or .

Corollary 2.14 (see ). is a maximal deductive system in the bounded Hilbert algebra if and only if for any ,

Lemma 2.15 (see ). The deductive system formed with the dense elements of a bounded Hilbert algebra , is the intersection of all maximal deductive systems in

Theorem 2.16 (see ). is an irreducible deductive system in the bounded Hilbert algebra if and only if for every there exists such that .

#### 3. Maximal and Irreducible Deductive Systems

A Hilbert algebra is said to be local if it has a single maximal deductive system. Using Lemma 2.15, we offer a necessary and sufficient condition for a bounded Hilbert algebra to be local.

Proposition 3.1. A bounded Hilbert algebra is local if and only if for , implies or .

Proof. If is local, from Lemma 2.15, is a maximal deductive system. Hence, if , then or as it is stated in Lemma 2.13. Thus, or . Conversely, let . Then, . It results that or and so, or . Since is maximal, from Lemma 2.15, it is the unique maximal deductive system in .

Lemma 3.2. Let be a local bounded Hilbert algebra with its single maximal deductive system. Then,

Proof. Let . Since is a proper deductive system, from Theorem 2.12, Hence,

Proposition 3.3 (see ). Every proper deductive system of a Hilbert algebra is the intersection of the irreducible deductive systems which contain .

Proposition 3.4. If is a proper deductive system in the bounded Hilbert algebra , then for

Proof. If we suppose that there exist such that from Corollary 2.8, . From Theorem 2.12, there exists a maximal deductive system such as But, since and Then, imply that , a contradiction.

Definition 3.5 (see ). Let be a subset of a bounded Hilbert algebra . is said to be -closed if, for each ,

Proposition 3.6. Let be a proper deductive system in a bounded Hilbert algebra Then, is maximal if and only if is a -closed subset.

Proof. If we consider to be a maximal deductive system and from Lemma 2.13, it results that . Conversely, let be a -closed set and let From Definition 3.5 it results that or To prove that is a maximal deductive system, we apply once again Lemma 2.13.

Proposition 3.7. Let be a bounded Hilbert algebra, a -closed subset of and a proper deductive system such that . Then, there exists a maximal deductive system such as and .

Proof. We consider the nonempty family It is easily verified that each chain in has an upper bound in Then, by Zorn's lemma, has a maximal element . All we have to do, furthermore, is to prove that is a maximal deductive system. If we presume that and , then and intersect as is a maximal element in . Let , , . Then, from Corollary 2.8, .
The relation implies that . Since , . Hence, . But, .
Since and , it results that From we obtain that This result contradicts the fact that since

Definition 3.8. Let be a subset of the bounded Hilbert algebra . We say that is a decreasing subset if for and implies that

Remark 3.9. It is easy to verify that if and only if for a decreasing -closed subset .

For an arbitrary element of a bounded Hilbert algebra we consider to be the deductive system generated by . If is a proper deductive system in , let us define the set as follows: Then, if and only if for each , as in Corollary 2.8. The last relation is equivalent to for all . We can see that otherwise .

Lemma 3.10. Let be a deductive system in a bounded Hilbert algebra Then, if and only if

Proof. If then, using the relation (8) of Theorem 2.4, . Conversely, let Since for each we have it results that for each Thus, .

Remark 3.11. If and intersect, and then . Hence, if we consider to be a proper deductive system, Thus, The equality holds if and only if is a maximal deductive system. Indeed, if and only if Hence, for all which is equivalent with Applying Corollary 2.14, this property is equivalent with the condition that is a maximal deductive system.

Proposition 3.12. defined as above is a decreasing -closed subset if is a proper deductive system in

Proof. Firstly, because we see that . Let and . Since and we get . Thus, . Now we consider . Then, and, from Proposition 3.4, .
But, from Theorem 2.4 (13), which proves that .

For a proper deductive system we define We obtain another deductive system which contains . Since , is also proper. If D is maximal, . Let be a maximal deductive system containing . If we get since Thus, Hence,

Remark 3.13. Let us consider a proper deductive system. Using the following chain of equivalences we obtain that

Proposition 3.14. For a proper deductive system in there is an irreducible deductive system such that

Proof. If we consider then Hence, else and Then and so Thus, , the family of all deductive systems which contain and are included in is non empty. It is easy to verify that every chain in has an upper bound in . From Zorn's Lemma, there exists , a maximal element in . To complete the proof, we show that is irreducible. Let's suppose that and , . Then, . Since we have . Let with . From Proposition 3.12, is -closed. Hence, since we obtain But and this contradicts the fact that

#### 4. The Spaces and

Let be a bounded Hilbert algebra. For each we define the set:

We will consider as a topological space with the class as a base.

Theorem 4.1 (see ). For each one has:(1) and ,(2), (3),(4),(5) where (6),(7).

Proposition 4.2. For , if and only if .

Proof. Let It means that for an arbitrary maximal deductive system , if and only if Assuming that , there exists an irreducible deductive system , as in Theorem 2.12, such that and Then, . Let be a maximal deductive system with Since then Yet, since is maximal and then From our supposition, we get that which contradicts the former result.
Thus, we obtain In a similar way, we prove that hence From the previous theorem the converse implication is also true.

It is known that a set of open subsets of a topological space is a base for the topology if and only if for each the set is a fundamental system of neighborhoods for

In our case, we have considered on the topology generated by the basis . Using this property, the following lemma results.

Lemma 4.3. Let Then, is a fundamental system of neighborhoods for

Theorem 4.4 (see ). Every Hilbert algebra is isomorphic with a subalgebra of the Hilbert algebra of all open subsets of the topological -space

In the proof of this theorem, is isomorphic to the class where for all This class is a subbase of the topological space .

Moreover, for all

The following lemma corresponds to Lemma 4.3.

Lemma 4.5. Let be an irreducible deductive system in the bounded Hilbert algebra . Then, is a fundamental system of neighborhoods for .

#### 5. Hilbert Algebras of Fractions

The Hilbert algebra of fractions with respect to a -closed subset is constructed in [4, 5] as follows.

Let be a bounded Hilbert algebra and be a -closed subset in . A congruence relation on is defined by , The corresponding quotient Hilbert algebra is denoted by and it is called the Hilbert algebra of fractions of with respect to the -closed subset . The congruence class of in will be denoted by

If is a deductive system in , it is easy to verify that is a deductive system in

Let be a maximal deductive system in . From Proposition 3.6, is a -closed subset and, in this case, we will denote the Hilbert algebra of fractions with respect to by

We know that a deductive system induces a congruence relation on defined by: , , as it is proved in . The resulting quotient Hilbert algebra will be denoted by and the congruence class of will be denoted by

Lemma 5.1. Let be a Hilbert algebra, a deductive system and a -closed subset in On we define the following relation: Then, is a congruence relation on

Proof. Let be the canonical morphism of Hilbert algebras. We easily see that is a -closed subset of Then, such that , such that
Hence, is exactly the congruence defined in the Hilbert algebra by means of the -closed subset

The quotient Hilbert algebra regarding the congruence relation will be denoted by and the corresponding congruence classes will be denoted by

For a maximal deductive system , let be the related -closed subset. Let be the canonical morphism defined by for each

Lemma 5.2. If is a deductive system in the bounded Hilbert algebra , then

Proof. Since for each maximal deductive system, we obtain one inclusion. To prove the converse inclusion, let us consider for each maximal deductive system . Then, There exists such that which means that there exist , , for all such that Since , and then From it results that for each . But each proper deductive system is included in a maximal one. Hence, Thus that is Finally, implies that

Lemma 5.3. Let , be two deductive systems in the Hilbert algebra . Then(i) if and only if for each maximal deductive system in (ii) if and only if for each maximal deductive system in (iii) if and only if for each maximal deductive system in (iv) if and only if for each maximal deductive system in

Proof. (i) We presume that for each maximal system Then, from Lemma 5.2,
results from the previous one.
(iii) We presume that for each maximal deductive system in Let Then, if and only if For we get that that is, This is equivalent to Hence, there exists such that So, there exists such that Since we obtain that
We suppose now that Let Hence, there exists with Then, , Since , we obtain that for Thus, there exists such that But is maximal and implies that Then, In a similar way, for It results that there exists such that Hence, These relations imply that showing that
(iv) From (ii) and (iii) it results that if and only if if and only if for each maximal deductive system in which is equivalent with if and only if for each maximal deductive system in

Proposition 5.4. If is a maximal deductive system in the Hilbert algebra the Hilbert algebra of fractions is local.

Proof. We show that is maximal using the result from Proposition 3.1. To do this, let's consider such that Then, there is such that Hence, But, from Corollary 2.14, and then The fact that is maximal implies either or We get that or Finally, or imply that or .

Lemma 5.5. In the bounded Hilbert algebra of fractions , if and only if there exists such as

Proof. If there exists with Hence, there exists and Conversely, if there is such that then Hence, which means that.Since we get.From these two relations, we obtain that Thus, and then, Hence,

Proposition 5.6. Let be a -closed subset in the bounded Hilbert algebra . One denotes the least decreasing subset generated by Then, is a -closed subset and it is the complement in of the union of all maximal systems which do not intersect

Proof. It is obvious that is a -closed subset. We denote the complement in of the union of all maximal systems which do not intersect by We shall prove that Let Then, for some If we suppose that there exists a maximal system which does not intersect and But then, , false. Conversely, if then for each maximal system with If , from Proposition 3.7, there exists a maximal deductive system such that and does not intersect false. Thus, intersects which means that

In , it is proved that the lattice of fractions verifies a property of universality. In the same way, we can easily deduce that the Hilbert algebras of fractions verify the following property of universality.

Proposition 5.7. Let be a bounded Hilbert algebra and a -closed subset in . Then, for each bounded Hilbert algebra and for any morphism of bounded Hilbert algebras with for all there exists an unique morphism such that which means that the following diagram is commutative: (5.7)

Proposition 5.8. Let and be two -closed subsets in the bounded Hilbert algebra , and we consider the morphism of Hilbert algebras defined by  for all The following statements are equivalent:(i) is bijective,(ii)(iii)for each maximal deductive system with ,

Proof. (i)(ii) Let . Since and is bijective, Hence, from Lemma 5.5,
(ii)(iii) Let be a maximal system with Let Since there exists with Hence, and we get
(iii)(ii) Let If then From Proposition 3.7, there exists a maximal deductive system with and Then, since we reach a contradiction.
(ii)(i) Let Thus, there exists such that Since for some we obtain that It results that: Then, Hence, and is bijective.

Proposition 5.9. Let be a bounded Hilbert algebra and a -closed subset in . If is local, there exists a maximal system such as and are isomorphic.

Proof. Let Because is the inverse image of the deductive system by means of the canonical map, it is a deductive system in In our case, is maximal in Hence, is also maximal.
Let us consider now the least decreasing -closed subset generated by . We prove that Let Then, if and only if which is equivalent to , as Lemma 3.2 states. Then, from Lemma 5.5, Finally, from Proposition 5.8,

Proposition 5.4 and Proposition 5.9 lead to the following result.

Theorem 5.10. A bounded Hilbert algebra of fractions is local if and only if there exists a maximal deductive system in such that and are isomorphic.

Now, we return to the previous subsets in and in where is a bounded Hilbert algebra. For each , one defines the sets:

Lemma 5.11. For each the following statements hold:(1) are decreasing -closed subsets. In fact, (2)

Proof. Let Then, for each Since is maximal, for each and so, Let now consider with . If there is a maximal deductive system such that Then, and so, false.
Since we have that As and is a decreasing -closed set, each is an element of Next, let's consider that and there exists such that Then, and, since we get false.
() since each maximal deductive system is an irreducible one. Let If we suppose that there is an irreducible deductive system with Then, and, finally, false.

For each , we define also the sets as the complement in of the union of all irreducible deductive systems

Lemma 5.12. With the previous notations,

Proof. Let Then, for all with Presuming that , from Lemma 5.11, Hence, there is an irreducible deductive system with and as Theorem 2.12 states. Since is a proper deductive system, it is included into a maximal one. Let be such a maximal deductive system.
From Theorem 2.4, for all From we obtain that , hence , for all . Then , and since we contradict the assumption that
For the converse inclusion, let us presume that and Then, and there exists with From Proposition 3.4, since each Hence, and then This contradiction ends the proof.

On the base of the topological space we can define a presheaf which means a contravariant functor from in the category of Hilbert algebras. We consider the open subsets Then, .

For let be the Hilbert algebra of fractions with respect to the -closed subset

Taking into account the result obtained in Proposition 5.7, we can consider the following commutative diagram: (5.11) where , are the canonical surjections and is defined by for all

Hence, each is put into correspondence with and the morphism commutes the following diagram: (5.12)

Lemma 5.13. For the set is directed.

Proof. For , using Proposition 3.4, Hence, since

We define now the -closed subset for each maximal deductive system . Since for each , there exists a morphism with as Proposition 5.7 states. It is obvious that each morphism is surjective.

Let us consider the family of morphisms For each such that we get Hence, for all with

Let be the direct limit of the directed system Since it verifies also a property of universality, there exists a unique morphism which commutes the diagram: (5.14)

Theorem 5.14. With the previous notations,

Proof. Let Then, there exists with which means that . Hence, for all We define by
We prove that the definition of is not depending on the choosing of To realize that, let such that We have to verify that Since as in Lemma 5.13, and then ,
The following diagram is a commutative one: (5.15)
Indeed, because the family of morphisms verifies the relation for each pairs such that In conclusion, is well defined.
Let with Then where is the equivalence class of 0 in the inductive limit. From the property of universality of algebras of fractions, there exists an unique morphism such that
From and we get that The relation implies that for each Since for all such that we have obtained two morphisms and which commute the diagram: (5.16)
Using the property of universality of the inductive limit of the directed system we get that Hence, is the inverse of the morphism and
To end the proof, we show that Let Then, where with Hence,
Conversely, since is maximal, for we have . From Lemma 5.11 we obtain

We now repeat the previous construction for irreducible deductive systems. From Lemmas 5.11 and 5.12 we know that for each and each , is a decreasing -closed subset in

Lemma 5.15. For the class is directed.

Proof. If we consider and be two elements of this class, Lemmas 5.11 and 5.13 imply that and ,

Now, let be an irreducible deductive system in We denote the union of all the -subsets when and with Then, is a decreasing -subset in

Theorem 5.16. With the previous notations, for an irreducible deductive system in the bounded Hilbert algebra

Proof. From Proposition 3.12, is a decreasing -closed subset.
Since is a decreasing -closed subset in , we obtain, as in the proof of Theorem 5.14, that Finally, we prove that
For , Then, Since we obtain that Conversely, let This implies that there exists an element such that Hence, But, As we get which is equivalent to Thus,