Hilbert Algebras of Fractions

Abstract

Let 𝐴 be a bounded Hilbert algebra and 𝑆 a ∨-closed subset of 𝐴. The Hilbert algebra of fractions 𝐴_𝑆 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 [1]. The theory of this kind of algebras has been developed by Diego in [2]. 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 [1]. 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 [3]. 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 Max(𝐴) and Ir(𝐴). 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 [6], we define a presheaf on the base of the topological space Max(𝐴) 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 𝐴 be a non empty set, → a binary operation on 𝐴 and 1∈𝐴.
A triplet (𝐴,→,1) is called Hilbert algebra if the following axioms hold, for each 𝑥,𝑦,𝑧∈𝐴:
(ℎ₁)𝑥→(𝑦→𝑥)=1,(ℎ₂)(𝑥→(𝑦→𝑧))→((𝑥→𝑦)→(𝑥→𝑧))=1,(ℎ₃)𝑥→𝑦=𝑦→𝑥=1 implies 𝑥=𝑦.

A Hilbert algebra 𝐴 becomes a poset by defining an order relation ≤ such that 𝑥≤𝑦 if and only if 𝑥→𝑦=1. 1 is the largest element of 𝐴 with respect to this order.

If the algebra 𝐴 has a smallest element, denoted by 0, it is called a bounded Hilbert algebra.

Definition 2.2. If 𝐴₁,𝐴₂ are bounded Hilbert algebras, 𝑓∶𝐴₁→𝐴₂ is said to be a morphism of Hilbert algebras if 𝑓(𝑥→𝑦)=𝑓(𝑥)→𝑓(𝑦) and 𝑓(0)=0.

Theorem 2.3 (see [2]). In a Hilbert algebra 𝐴 the following relations hold for each 𝑥,𝑦,𝑧∈𝐴:(1)𝑥≤𝑦→𝑥, (2)𝑥→1=1,(3)𝑥→(𝑦→𝑧)=(𝑥→𝑦)→(𝑥→𝑧),(4)(𝑥→𝑦)→((𝑦→𝑥)→𝑥)=(𝑦→𝑥)→((𝑥→𝑦)→𝑦),(5)𝑥→(𝑦→𝑧)=𝑦→(𝑥→𝑧),(6)𝑥≤(𝑥→𝑦)→𝑦;((𝑥→𝑦)→𝑦)→𝑦=𝑥→𝑦, (7)1→𝑥=𝑥, (8)𝑥→𝑦≤(𝑦→𝑧)→(𝑥→𝑧),(9)if 𝑥≤𝑦, then 𝑧→𝑥≤𝑧→𝑦 and 𝑦→𝑧≤𝑥→𝑧.We define 𝑥⊻𝑦=𝑥^∗→𝑦 and 𝑥⊼𝑦=(𝑥→𝑦^∗)^∗ where 𝑥^∗=𝑥→0.

Theorem 2.4 (see [3]). In a bounded Hilbert algebra 𝐴, the following relations hold for each of the elements 𝑥,𝑦,𝑧∈𝐴:(1)0^∗=1 and 1^∗=0,(2)𝑥→𝑦^∗=𝑦→𝑥^∗,(3)𝑥→𝑥^∗=𝑥^∗ and 𝑥^∗→𝑥=𝑥^∗∗,(4)𝑥→𝑦≤𝑦^∗→𝑥^∗,(5)𝑥≤𝑦 implies that 𝑦^∗≤𝑥^∗,(6)(𝑥→𝑦)^∗∗=𝑥→𝑦^∗∗=𝑥^∗∗→𝑦^∗∗,(7)𝑥≤𝑥⊻𝑦; 𝑦≤𝑥⊻𝑦, (8)𝑥⊻𝑥=𝑥^∗∗; 𝑥⊻1=1; 1⊻𝑥=1; 𝑥⊻𝑥^∗=1,(9)𝑥⊻(𝑦→𝑧)=(𝑥⊻𝑦)→(𝑥⊻𝑧),(10)𝑥⊻(𝑦⊻𝑧)=𝑦⊻(𝑥⊻𝑧)=(𝑥⊻𝑦)⊻𝑧,(11)𝑥⊼𝑦=𝑦⊼𝑥,(12)𝑥⊼𝑦≤𝑥^∗∗, 𝑥⊼𝑦≤𝑦^∗∗.(13)𝑥^∗⊼𝑦^∗=(𝑥⊻𝑦)^∗.

Definition 2.5. A subset 𝐷 of a Hilbert algebra 𝐴 is called a deductive system if 1∈𝐷 and from 𝑥,𝑥→𝑦∈𝐷 it results that 𝑦∈𝐷.

In a Hilbert algebra 𝐴, for 𝑥,𝑦∈𝐴, Jun defines in [7] the deductive system 𝐴(𝑥,𝑦)={𝑧∈𝐴∣𝑥→(𝑦→𝑧)=1}. Using this notion, he gives the following characterization of deductive systems.

Theorem 2.6 (see [7]). Let 𝐷 be a nonempty subset of a Hilbert algebra. Then, 𝐷 is a deductive system of 𝐴 if and only if for any 𝑥,𝑦∈𝐷,𝐴(𝑥,𝑦)⊆𝐷.

We denote the set of all deductive systems in 𝐴 with 𝐷𝑠(𝐴).

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

Theorem 2.7 (see [1]). Let 𝐴 be a Hilbert algebra and 𝑆 a subset of 𝐴. The deductive system generated by 𝑆 is the set of the elements 𝑥∈𝐴 for which there exist 𝑎₁,…,𝑎_𝑛∈𝑆 such that 𝑎₁→(𝑎₂→(⋯(𝑎_𝑛→𝑥)⋯))=1.

In particular, if we consider the set 𝑆={𝑎}, the deductive system [𝑎)={𝑥∈𝐴∣𝑎≤𝑥} (2.1) is called the deductive system generated by 𝑎.

In [8] it is proved that 𝐷𝑠(𝐴) is a Heyting algebra with respect to the following operations:

𝐷₁⋀𝐷₂=𝐷₁∩𝐷₂,𝐷₁∨𝐷₂=[𝐷₁∪𝐷₂),𝐷₁→𝐷₂={𝑎∈𝐴∣[𝑎)∩𝐷₁⊆𝐷₂}.

Corollary 2.8 (see [9]). If 𝐴 is a Hilbert algebra, 𝐷∈𝐷𝑠(𝐴), and 𝑎∈𝐴, then [𝑎)∨𝐷={𝑥∈𝐴∣𝑎→𝑥∈𝐷}. (2.2)

In [10] is defined the concept of ideals in a Hilbert algebra as follows.

Definition 2.9 (see [10]). A nonempty subset 𝐼 of a Hilbert algebra 𝐴 is called an ideal of 𝐴 if:(1)1∈𝐼, (2)𝑥→𝑦∈𝐼, for all 𝑥∈𝐴,𝑦∈𝐼, (3)(𝑦₂→(𝑦₁→𝑥))→𝑥∈𝐼, for all 𝑥∈𝐴 and 𝑦₁,𝑦₂∈𝐼.

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 [12] 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 [13] it is proved much more: 𝐷𝑠(𝐴) is an algebraic lattice which is distributive and isomorphic with Con(𝐴), the lattice of all congruences on 𝐴.

Definition 2.10 (see [1]). Let 𝐴 be a Hilbert algebra and 𝐷 a proper deductive system. 𝐷 is called irreducible if for any two proper deductive systems 𝐷₁,𝐷₂ such that 𝐷=𝐷₁∩𝐷₂ either 𝐷=𝐷₁ or 𝐷=𝐷₂.

Definition 2.11 (see [1]). 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 𝐴.

For a Hilbert algebra 𝐴 we will denote the set of all maximal deductive systems of 𝐴 by Max(𝐴) and the set of all irreducible deductive systems of 𝐴 with Ir(𝐴). According to their definitions, it is obvious that Max(𝐴)⊆Ir(𝐴).

Theorem 2.12 (see [1]). Let 𝐴 be a Hilbert algebra.(i)If 𝐷 is a proper deductive system and 𝑎∈𝐴⧵𝐷, there exists an irreducible deductive system 𝐼 such that 𝐷⊆𝐼 and 𝑎∈𝐴⧵𝐼.(ii)If 𝑎≰0𝑥00092𝑏, there exists an irreducible deductive system 𝐼 such that 𝑎∈𝐼 and 𝑏∈𝐴⧵𝐼.(iii)If 𝐴 is bounded, each proper deductive system is included in a maximal one.

Lemma 2.13 (see [3]). Let 𝐴 be a bounded Hilbert algebra. The following statements are equivalent:(i)𝑀 is a maximal deductive system in 𝐴.(ii)For 𝑥,𝑦∈𝐴 and 𝑥⊻𝑦∈𝑀, either 𝑥∈𝑀 or 𝑦∈𝑀.

Corollary 2.14 (see [3]). 𝑀 is a maximal deductive system in the bounded Hilbert algebra 𝐴 if and only if for any 𝑥∈𝐴⧵𝑀, 𝑥^∗∈𝑀.

Lemma 2.15 (see [3]). The deductive system 𝒟(𝐴)={𝑥∈𝐴∣𝑥^∗=0}, formed with the dense elements of a bounded Hilbert algebra 𝐴, is the intersection of all maximal deductive systems in 𝐴.

Theorem 2.16 (see [2]). 𝐼 is an irreducible deductive system in the bounded Hilbert algebra 𝐴 if and only if for every 𝑥,𝑦∉𝐼 there exists 𝑧∉𝐼 such that 𝑥,𝑦≤𝑧.

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 𝐴 is local if and only if for 𝑥,𝑦∈𝐴, 𝑥⊻𝑦=1 implies 𝑥^∗∗=1 or 𝑦^∗∗=1.

Proof. If 𝐴 is local, from Lemma 2.15, 𝒟(𝐴) is a maximal deductive system. Hence, if 𝑥⊻𝑦=1∈𝒟(𝐴), then 𝑥∈𝒟(𝐴) or 𝑦∈𝒟(𝐴) as it is stated in Lemma 2.13. Thus, 𝑥^∗∗=1 or 𝑦^∗∗=1. Conversely, let 𝑥⊻𝑦∈𝒟(𝐴). Then, 1=(𝑥⊻𝑦)^∗∗=𝑥⊻𝑦^∗∗. It results that 𝑥^∗∗=1 or 𝑦^∗∗=1 and so, 𝑥∈𝒟(𝐴) or 𝑦∈𝒟(𝐴). Since 𝒟(𝐴) is maximal, from Lemma 2.15, it is the unique maximal deductive system in 𝐴.

Lemma 3.2. Let 𝐴 be a local bounded Hilbert algebra with 𝒟(𝐴) its single maximal deductive system. Then, 𝒟(𝐴)=𝐴⧵{0}.

Proof. Let 𝑥∈𝐴⧵{0}. Since [𝑥) is a proper deductive system, from Theorem 2.12, [𝑥)⊆𝒟(𝐴). Hence, 𝑥∈𝒟(𝐴).

Proposition 3.3 (see [9]). Every proper deductive system 𝐷 of a Hilbert algebra is the intersection of the irreducible deductive systems which contain 𝐷.

Proposition 3.4. If 𝐷 is a proper deductive system in the bounded Hilbert algebra 𝐴, then 𝑥⊼𝑦∈𝐷, for 𝑥,𝑦∈𝐷.

Proof. If we suppose that there exist 𝑥,𝑦∈𝐷 such that 𝑥⊼𝑦=(𝑥→𝑦^∗)^∗∉𝐷, from Corollary 2.8, 0∉𝐷∨[(𝑥→𝑦^∗)). From Theorem 2.12, there exists a maximal deductive system 𝑀 such as 𝐷∨[(𝑥→𝑦^∗))⊆𝑀. But, 𝑦^∗∈𝑀 since 𝑥∈𝑀 and 𝑥→𝑦^∗∈𝑀. Then, 𝑦^∗,𝑦∈𝑀 imply that 𝑀=𝐴, a contradiction.

Definition 3.5 (see [5]). Let 𝑆 be a subset of a bounded Hilbert algebra 𝐴. 𝑆 is said to be ⊻-closed if, for each 𝑥,𝑦∈𝑆, 𝑥⊻𝑦∈𝑆.

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

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

Proposition 3.7. Let 𝐴 be a bounded Hilbert algebra, 𝑆 a ⊻-closed subset of 𝐴 and 𝐷 a proper deductive system such that 𝑆∩𝐷=∅. Then, there exists a maximal deductive system 𝑀 such as 𝐷⊆𝑀 and 𝑀∩𝑆=∅.

Proof. We consider the nonempty family 𝐓={𝐹∣𝐹deductivesystem,𝐷⊆𝐹,𝐹∩𝑆=∅}. (3.1) It is easily verified that each chain in 𝐓 has an upper bound in 𝐓. Then, by Zorn's lemma, 𝐓 has a maximal element 𝑀. All we have to do, furthermore, is to prove that 𝑀 is a maximal deductive system. If we presume that 𝑥⊻𝑦∈𝑀 and 𝑥,𝑦∈𝐴⧵𝑀, then 𝑀∨[𝑥) and 𝑀∨[𝑦) intersect 𝑆, as 𝑀 is a maximal element in 𝐓. Let 𝑠,𝑡∈𝑆, 𝑠∈𝑀∨[𝑥), 𝑡∈𝑀∨[𝑦). Then, from Corollary 2.8, 𝑥→𝑠,𝑦→𝑡∈𝑀.
The relation 𝑦→𝑡≤𝑥⊻(𝑦→𝑡) implies that 𝑥⊻(𝑦→𝑡)=𝑥⊻𝑦→𝑥⊻𝑡∈𝑀. Since 𝑥⊻𝑦∈𝑀, 𝑥⊻𝑡∈𝑀. Hence, 𝑠⊻(𝑥⊻𝑡)∈𝑀. But, 𝑠⊻(𝑥⊻𝑡)=𝑠^∗→(𝑥^∗→𝑡)=(𝑠^∗→𝑥^∗)→(𝑠^∗→𝑡)=(𝑥→𝑠^∗∗)→𝑠⊻𝑡=(𝑥→𝑠)^∗∗→𝑠⊻𝑡.
Since 𝑥→𝑠≤(𝑥→𝑠)^∗∗ and 𝑥→𝑠∈𝑀, it results that (𝑥→𝑠)^∗∗∈𝑀. From (𝑥→𝑠)^∗∗→(𝑠⊻𝑡)∈𝑀 we obtain that 𝑠⊻𝑡∈𝑀. This result contradicts the fact that 𝑀∩𝑆=∅ since 𝑠⊻𝑡∈𝑆.

Definition 3.8. Let 𝑆 be a subset of the bounded Hilbert algebra 𝐴. We say that 𝑆 is a decreasing subset if for 𝑠∈𝑆 and 𝑥∈𝐴,𝑥≤𝑠 implies that 𝑥∈𝑆.

Remark 3.9. It is easy to verify that 𝑥⊻𝑦∈𝑆 if and only if 𝑥,𝑦∈𝑆 for a decreasing ⊻-closed subset 𝑆.

For an arbitrary element 𝑥 of a bounded Hilbert algebra 𝐴 we consider [𝑥^∗∗) to be the deductive system generated by 𝑥^∗∗. If 𝐷 is a proper deductive system in 𝐴, let us define the set 𝐷 as follows: 𝐷={𝑥∈𝐴∣𝐷∨[𝑥^∗∗)=𝐴}. (3.2) Then, 𝑥∈𝐷 if and only if for each 𝑎∈𝐴, 𝑥^∗∗→𝑎∈𝐷, as in Corollary 2.8. The last relation is equivalent to 𝑥^∗⊻𝑎∈𝐷, for all 𝑎∈𝐴. We can see that 1∉𝐷, otherwise 𝐷=𝐴.

Lemma 3.10. Let 𝐷 be a deductive system in a bounded Hilbert algebra 𝐴. Then, 𝑥∈𝐷 if and only if 𝑥^∗∈𝐷.

Proof. If 𝑥∈𝐷 then, using the relation (8) of Theorem 2.4, 𝑥^∗⊻𝑥^∗=𝑥^∗∈𝐷. Conversely, let 𝑥^∗∈𝐷. Since for each 𝑎∈𝐴 we have 𝑥^∗≤(𝑥^∗⊻𝑎), it results that 𝑥^∗⊻𝑎∈𝐷, for each 𝑎∈𝐴. Thus, 𝑥∈𝐷.

Remark 3.11. If 𝐷 and 𝐷 intersect, 0∈𝐷 and then 𝐴=𝐷. Hence, if we consider 𝐷 to be a proper deductive system, 𝐷∩𝐷=∅. Thus, 𝐷⊆𝐴⧵𝐷. The equality holds if and only if 𝐷 is a maximal deductive system. Indeed, 𝐷=𝐴⧵𝐷 if and only if 𝐴⧵𝐷⊆𝐷. Hence, for all 𝑥∉𝐷,𝑥∈𝐷 which is equivalent with 𝑥^∗∈𝐷. Applying Corollary 2.14, this property is equivalent with the condition that 𝐷 is a maximal deductive system.

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

Proof. Firstly, because 1∈𝐷, we see that 0∈𝐷. Let 𝑥≤𝑦 and 𝑦∈𝐷. Since 𝑦^∗≤𝑥^∗ and 𝑦^∗∈𝐷, we get 𝑥^∗∈𝐷. Thus, 𝑥∈𝐷. Now we consider 𝑥,𝑦∈𝐷. Then, 𝑥^∗,𝑦^∗∈𝐷 and, from Proposition 3.4, 𝑥^∗⊼𝑦^∗∈𝐷.
But, from Theorem 2.4 (13), 𝑥^∗⊼𝑦^∗=(𝑥⊻𝑦)^∗∈𝐷 which proves that 𝑥⊻𝑦∈𝐷.

For a proper deductive system 𝐷 we define 𝐷₀={𝑥∈𝐴∣𝑥^∗∗∈𝐷}. (3.3) We obtain another deductive system which contains 𝐷. Since 0∉𝐷, 𝐷₀ is also proper. If D is maximal, 𝐷=𝐷₀. Let 𝑀 be a maximal deductive system containing 𝐷. If 𝑥∈𝐷₀, we get 𝑥^∗∗∈𝑀, since 𝑥^∗∗∈𝐷⊆𝑀. Thus, 𝑥∈𝑀. Hence, 𝐷₀⊆𝑀.

Remark 3.13. Let us consider 𝐷 a proper deductive system. Using the following chain of equivalences 𝑥∈𝐷⇔𝑥^∗∈𝐷⇔𝑥^∗∈𝐷₀⇔𝑥∈𝐷₀, (3.4) we obtain that 𝐷=𝐷₀.

Proposition 3.14. For a proper deductive system 𝐷 in 𝐴 there is an irreducible deductive system 𝐽 such that 𝐷⊆𝐽⊆𝐴⧵𝐷.

Proof. If we consider 𝑥∈𝐷₀, then 𝑥^∗∗∈𝐷. Hence, 𝑥^∗∉𝐷, else 0∈𝐷 and 𝐷=𝐴. Then 𝑥∉𝐷 and so 𝐷⊆𝐷₀⊆𝐴⧵𝐷. Thus, 𝐅, the family of all deductive systems which contain 𝐷 and are included in 𝐴⧵𝐷, is non empty. It is easy to verify that every chain in 𝐅 has an upper bound in 𝐅. From Zorn's Lemma, there exists 𝐽, a maximal element in 𝐅. To complete the proof, we show that 𝐽 is irreducible. Let's suppose that 𝐽=𝐽₁∩𝐽₂ and 𝐽≠𝐽₁, 𝐽≠𝐽₂. Then, 𝐽₁,𝐽₂∉𝐅. Since 𝐷⊆𝐽⊆𝐽₁,𝐽₂ we have 𝐽₁,𝐽₂⊆𝐴̸⧵𝐷. Let 𝑎∈𝐽₁,𝑏∈𝐽₂ with 𝑎,𝑏∉𝐴⧵𝐷. From Proposition 3.12, 𝐷 is ⊻-closed. Hence, since 𝑎,𝑏∈𝐷 we obtain 𝑎⊻𝑏∈𝐷. But 𝑎⊻𝑏∈𝐽₁∩𝐽₂=𝐽 and this contradicts the fact that 𝐽∈𝐅.

4. The Spaces 𝐌𝐚𝐱(𝐴) and 𝐈𝐫(𝐴)

Let 𝐴 be a bounded Hilbert algebra. For each 𝑥∈𝐴, we define the set: 𝑟(𝑥)={𝑀∈Max(𝐴)∣𝑥∈𝑀}. (4.1)

We will consider Max(𝐴) as a topological space with the class {𝑟(𝑥)∣𝑥∈𝐴} as a base.

Theorem 4.1 (see [3]). For each 𝑥,𝑦∈𝐴 one has:(1)𝑟(0)=∅ and 𝑟(1)=Max(𝐴),(2)𝑥≤𝑦⇒𝑟(𝑥)⊆𝑟(𝑦), (3)𝑟(𝑥^∗)=Int(Max(𝐴)⧵𝑟(𝑥)),(4)𝑟(𝑥^∗∗)=𝑟(𝑥),(5)𝑟(𝑥→𝑦)=𝑟(𝑥)→𝑟(𝑦) where 𝑟(𝑥)→𝑟(𝑦)=Int((Max(𝐴)⧵𝑟(𝑥))∪𝑟(𝑦)),(6)𝑟(𝑥)∩𝑟(𝑦)=𝑟(𝑥⊼𝑦),(7)𝑟(𝑥)∪𝑟(𝑦)=𝑟(𝑥⊻𝑦).

Proposition 4.2. For 𝑥,𝑦∈𝐴, 𝑟(𝑥)=𝑟(𝑦) if and only if 𝑥^∗∗=𝑦^∗∗.

Proof. Let 𝑟(𝑥)=𝑟(𝑦). It means that for an arbitrary maximal deductive system 𝑀, 𝑥∈𝑀 if and only if 𝑦∈𝑀. Assuming that 𝑥^∗∗0𝑥00092≰𝑦^∗∗, there exists an irreducible deductive system 𝐼, as in Theorem 2.12, such that 𝑥^∗∗∈𝐼 and 𝑦^∗∗∉𝐼. Then, 0∉𝐼∨[𝑦^∗). Let 𝑀₁ be a maximal deductive system with 𝐼∨[𝑦^∗)⊆𝑀₁. Since 𝑦^∗∈𝑀₁, then 𝑦∉𝑀₁. Yet, since 𝑀₁ is maximal and 𝑥^∗∗∈𝐼⊆𝑀₁, then 𝑥∈𝑀₁. From our supposition, 𝑟(𝑥)=𝑟(𝑦), we get that 𝑦∈𝑀₁ which contradicts the former result.
Thus, we obtain 𝑥^∗∗≤𝑦^∗∗. In a similar way, we prove that 𝑦^∗∗≤𝑥^∗∗, hence 𝑥^∗∗=𝑦^∗∗. From the previous theorem the converse implication is also true.

It is known that a set 𝐁 of open subsets of a topological space (𝑋,𝜏) is a base for the topology 𝜏 if and only if for each 𝑥∈𝑋, the set 𝐵_𝑥={𝐵∣𝐵∈𝐁,𝑥∈𝐵} is a fundamental system of neighborhoods for 𝑥.

In our case, we have considered on Max(𝐴) the topology generated by the basis {𝑟(𝑥)∣𝑥∈𝐴}. Using this property, the following lemma results.

Lemma 4.3. Let 𝑀∈Max(𝐴). Then, 𝑉_𝑀={𝑟(𝑥)∣𝑥∈𝑀} is a fundamental system of neighborhoods for 𝑀.

Theorem 4.4 (see [9]). Every Hilbert algebra 𝐴 is isomorphic with a subalgebra of the Hilbert algebra of all open subsets of the topological 𝑇₀-space Ir(𝐴).

In the proof of this theorem, 𝐴 is isomorphic to the class {𝑠(𝑥)∣𝑥∈𝐴} where 𝑠(𝑥)={𝐼∈Ir(𝐴)∣𝑥∈𝐼}, for all 𝑥∈𝐴. This class is a subbase of the topological space Ir(𝐴).

Moreover, 𝑠(𝑥→𝑦)=Int((Ir(𝐴)⧵𝑠(𝑥))∪𝑠(𝑦)) for all 𝑥,𝑦∈𝐴.

The following lemma corresponds to Lemma 4.3.

Lemma 4.5. Let 𝐼 be an irreducible deductive system in the bounded Hilbert algebra 𝐴. Then, 𝑉_𝐼={^𝑛_𝑖=1𝑠(𝑥_𝑖)∣𝑥_𝑖∈𝐼,1≤𝑖≤𝑛,𝑛≥1} (4.2) is a fundamental system of neighborhoods for 𝐼.

5. Hilbert Algebras of Fractions

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

Let 𝐴 be a bounded Hilbert algebra and 𝑆 be a ⊻-closed subset in 𝐴. A congruence relation 𝜃_𝑆 on 𝐴 is defined by (𝑥,𝑦)∈𝜃_𝑆⇔∃𝑠∈𝑆, 𝑠⊻𝑥=𝑠⊻𝑦. The corresponding quotient Hilbert algebra 𝐴/_𝜃𝑆 is denoted by 𝐴_𝑆 and it is called the Hilbert algebra of fractions of 𝐴 with respect to the ⊻-closed subset 𝑆. The congruence class of 𝑥 in 𝐴_𝑆 will be denoted by [𝑥]_𝑆.

If 𝐷 is a deductive system in 𝐴, it is easy to verify that 𝐷_𝑆={[𝑥]_𝑆∣𝑥∈𝐷} is a deductive system in 𝐴_𝑆.

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

We know that a deductive system 𝐷 induces a congruence relation 𝜃_𝐷 on 𝐴 defined by: (𝑥,𝑦)∈𝜃_𝐷⇔𝑥→𝑦, 𝑦→𝑥∈𝐷, as it is proved in [1]. The resulting quotient Hilbert algebra will be denoted by 𝐴/_𝐷 and the congruence class of 𝑥 will be denoted by 𝑥/_𝐷.

Lemma 5.1. Let 𝐴 be a Hilbert algebra, 𝐷 a deductive system and 𝑆 a ⊻-closed subset in 𝐴. On 𝐴/_𝐷 we define the following relation: (𝑥/_𝐷,𝑦/_𝐷)∈𝜃^′_𝑆⇔∃𝑠∈𝑆,(𝑠⊻𝑥)/_𝐷=(𝑠⊻𝑦)/_𝐷. (5.1) Then, 𝜃^′_𝑆 is a congruence relation on 𝐴/_𝐷.

Proof. Let 𝑝_𝐷∶𝐴→𝐴/_𝐷 be the canonical morphism of Hilbert algebras. We easily see that 𝑆_𝐷=𝑝_𝐷(𝑆)={𝑝_𝐷(𝑠)∣𝑠∈𝑆} is a ⊻-closed subset of 𝐴/_𝐷. Then, (𝑥/_𝐷,𝑦/_𝐷)∈𝜃_𝑆𝐷⇔∃𝑡∈𝑆_𝐷 such that 𝑡⊻𝑥/_𝐷=𝑡⊻𝑦/_𝐷⇔∃𝑠∈𝑆, 𝑡=𝑠/_𝐷, such that (𝑠⊻𝑥)/_𝐷=𝑡⊻𝑥/_𝐷=𝑡⊻𝑦/_𝐷=(𝑠⊻𝑦)/_𝐷⇔(𝑥/_𝐷,𝑦/_𝐷)∈𝜃^′_𝑆.
Hence, 𝜃^′_𝑆 is exactly the congruence defined in the Hilbert algebra 𝐴/_𝐷 by means of the ⊻-closed subset 𝑆_𝐷=𝑝_𝐷(𝑆).

The quotient Hilbert algebra regarding the congruence relation 𝜃^′_𝑆 will be denoted by (𝐴/_𝐷)_𝑆 and the corresponding congruence classes will be denoted by [𝑥/_𝐷]_𝑆.

For a maximal deductive system 𝑀, let 𝑆=𝐴⧵𝑀 be the related ⊻-closed subset. Let 𝑝_𝑀∶𝐴→𝐴_𝑀 be the canonical morphism defined by 𝑝_𝑀(𝑥)=[𝑥]_𝑀 for each 𝑥∈𝐴.

Lemma 5.2. If 𝐷 is a deductive system in the bounded Hilbert algebra 𝐴, then 𝐷=_{𝑀∈Max(𝐴)}𝑝_𝑀⁻¹(𝐷_𝑀). (5.2)

Proof. Since 𝐷⊆𝑝_𝑀⁻¹(𝐷_𝑀) for each maximal deductive system, we obtain one inclusion. To prove the converse inclusion, let us consider 𝑥∈𝑝_𝑀⁻¹(𝐷_𝑀), for each maximal deductive system 𝑀. Then, 𝑝_𝑀(𝑥)∈⋂_{𝑀∈Max(𝐴)}𝐷_𝑀. There exists 𝑎∈𝐷 such that [𝑥]_𝑀=[𝑎]_𝑀 which means that there exist 𝑎∈𝐷, 𝑡∉𝑀, for all 𝑀, such that 𝑡⊻𝑥=𝑡⊻𝑎. Since 𝑎∈𝐷, 𝑡⊻𝑎∈𝐷 and then 𝑡⊻𝑥∈𝐷. From 𝑡∉𝑀 it results that [𝑡)∨𝐷={𝑦∈𝐴∣𝑡→𝑦∈𝐷}⊆𝑀̸, for each 𝑀. But each proper deductive system is included in a maximal one. Hence, [𝑡)∨𝐷=𝐴. Thus 0∈[𝑡)∨𝐷, that is 𝑡^∗=𝑡→0∈𝐷. Finally, 𝑡⊻𝑥=𝑡^∗→𝑥∈𝐷 implies that 𝑥∈𝐷.

Lemma 5.3. Let 𝐷₁, 𝐷₂ be two deductive systems in the Hilbert algebra 𝐴. Then(i)𝐷₁⊆𝐷₂ if and only if (𝐷₁)_𝑀⊆(𝐷₂)_𝑀 for each maximal deductive system 𝑀 in 𝐴,(ii)𝐷₁=𝐷₂ if and only if (𝐷₁)_𝑀=(𝐷₂)_𝑀 for each maximal deductive system 𝑀 in 𝐴,(iii)(𝐷₁)_𝑀=(𝐷₂)_𝑀 if and only if (𝐴/_𝐷1)_𝑀≃(𝐴/_𝐷2)_𝑀 for each maximal deductive system 𝑀 in 𝐴,(iv)𝐴/_𝐷1≃𝐴/_𝐷2 if and only if (𝐴/_𝐷1)_𝑀≃(𝐴/_𝐷2)_𝑀 for each maximal deductive system 𝑀 in 𝐴.

Proof. (i) We presume that (𝐷₁)_𝑀⊆(𝐷₂)_𝑀 for each maximal system 𝑀. Then, from Lemma 5.2, 𝐷₁=_{𝑀∈Max(𝐴)}𝑝_𝑀⁻¹((𝐷₁)_𝑀)⊆_{𝑀∈Max(𝐴)}𝑝_𝑀⁻¹((𝐷₂)_𝑀)=𝐷₂. (5.3)
(ii) results from the previous one.
(iii) We presume that (𝐴/_𝐷1)_𝑀≃(𝐴/_𝐷2)_𝑀 for each maximal deductive system 𝑀 in 𝐴. Let 𝑥,𝑦∈𝐴. Then, [𝑥/_𝐷1]_𝑀=[𝑦/_𝐷1]_𝑀 if and only if [𝑥/_𝐷2]_𝑀=[𝑦/_𝐷2]_𝑀. For 𝑥∈𝐷₁ we get that 𝑥/_𝐷1=1/_𝐷1, that is, [𝑥/_𝐷1]_𝑀=[1/_𝐷1]_𝑀. This is equivalent to [𝑥/_𝐷2]_𝑀=[1/_𝐷2]_𝑀. Hence, there exists 𝑡∉𝑀 such that (𝑡⊻𝑥)/_𝐷2=(𝑡⊻1)/_𝐷2=1/_𝐷2. So, there exists 𝑡∉𝑀 such that 𝑡⊻𝑥∈𝐷₂. Since 𝑡⊻𝑥=𝑡⊻(𝑡⊻𝑥) we obtain that [𝑥]_𝑀=[(𝑡⊻𝑥)]_𝑀∈(𝐷₂)_𝑀.
We suppose now that (𝐷₁)_𝑀=(𝐷₂)_𝑀. Let [𝑥/_𝐷1]_𝑀=[𝑦/_𝐷1]_𝑀. Hence, there exists 𝑡∉𝑀 with (𝑡⊻𝑥)/_𝐷1=(𝑡⊻𝑦)/_𝐷1. Then, 𝑡⊻(𝑥→𝑦), 𝑡⊻(𝑦→𝑥)∈𝐷₁. Since (𝐷₁)_𝑀=(𝐷₂)_𝑀, we obtain that [𝑡⊻(𝑥→𝑦)]_𝑀=[𝑢]_𝑀 for 𝑢∈𝐷₂. Thus, there exists 𝑠∉𝑀 such that 𝑠⊻𝑡⊻(𝑥→𝑦)=𝑠⊻𝑢. But 𝑀 is maximal and 𝑠,𝑡∉𝑀 implies that 𝑠⊻𝑡∉𝑀. Then, (𝑠⊻𝑡)⊻(𝑥→𝑦)=𝑠⊻𝑢∈𝐷₂. In a similar way, [𝑡⊻(𝑦→𝑥)]_𝑀=[𝑣]_𝑀 for 𝑣∈𝐷₂. It results that there exists 𝑠₁∉𝑀 such that (𝑠₁⊻𝑡)⊻(𝑦→𝑥)=𝑠₁⊻𝑣∈𝐷₂. Hence, 𝑠₁⊻𝑠⊻𝑡∉𝑀,(𝑠₁⊻𝑠⊻𝑡)⊻(𝑦→𝑥),(𝑠₁⊻𝑠⊻𝑡)⊻(𝑥→𝑦)∈𝐷₂. (5.4) These relations imply that ((𝑠₁⊻𝑠⊻𝑡)⊻𝑥)/_𝐷2=((𝑠₁⊻𝑠⊻𝑡)⊻𝑦)/_𝐷2, showing that [𝑥/_𝐷2]_𝑀=[𝑦/_𝐷2]_𝑀.
(iv) From (ii) and (iii) it results that 𝐴/_𝐷1≃𝐴/_𝐷2 if and only if 𝐷₁=𝐷₂ if and only if (𝐷₁)_𝑀=(𝐷₂)_𝑀 for each maximal deductive system 𝑀 in 𝐴 which is equivalent with 𝐴/_𝐷1≃𝐴/_𝐷2 if and only if (𝐴/_𝐷1)_𝑀≃(𝐴/_𝐷2)_𝑀 for each maximal deductive system 𝑀 in 𝐴.

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

Proof. We show that 𝒟(𝐴_𝑀) is maximal using the result from Proposition 3.1. To do this, let's consider 𝑥,𝑦∈𝐴 such that [𝑥⊻𝑦]_𝑀=[1]_𝑀. Then, there is 𝑡∉𝑀 such that 𝑡⊻(𝑥⊻𝑦)=𝑡⊻1=1. Hence, 𝑡^∗≤𝑥⊻𝑦. But, from Corollary 2.14, 𝑡^∗∈𝑀 and then 𝑥⊻𝑦∈𝑀. The fact that 𝑀 is maximal implies either 𝑥∈𝑀 or 𝑦∈𝑀. We get that 𝑥^∗∉𝑀 or 𝑦^∗∉𝑀. Finally, 𝑥^∗⊻𝑥^∗∗=𝑥^∗⊻1 or 𝑦^∗⊻𝑦^∗∗=𝑦^∗⊻1 imply that [𝑥^∗∗]_𝑀=[1]_𝑀 or [𝑦^∗∗]_𝑀=[1]_𝑀.

Lemma 5.5. In the bounded Hilbert algebra of fractions 𝐴_𝑆, [𝑥]_𝑆=[0]_𝑆 if and only if there exists 𝑠∈𝑆 such as 𝑥≤𝑠.

Proof. If [𝑥]_𝑆=[0]_𝑆 there exists 𝑡∈𝑆 with 𝑡⊻𝑥=𝑡⊻0=𝑡^∗∗=𝑡⊻𝑡∈𝑆. Hence, there exists 𝑠=𝑡⊻𝑥∈𝑆 and 𝑥≤𝑠. Conversely, if there is 𝑠∈𝑆 such that 𝑥≤𝑠, then 𝑠⊻𝑥≤𝑠⊻𝑠=𝑠^∗∗. Hence, (𝑠⊻𝑥)^∗∗≤𝑠^∗∗ which means that(1)𝑠^∗≤(𝑠⊻𝑥)^∗.Since 𝑠≤𝑠⊻𝑥, we get(2)(𝑠⊻𝑥)^∗≤𝑠^∗.From these two relations, we obtain that (𝑠⊻𝑥)^∗=𝑠^∗. Thus, 𝑠⊻𝑥=(𝑠⊻𝑥)⊻𝑥=𝑠⊻(𝑥⊻𝑥)=𝑠⊻𝑥^∗∗ (5.5) and then, 𝑠⊻𝑥=𝑠⊻𝑥^∗∗=(𝑠⊻𝑥)^∗∗=𝑠^∗∗=𝑠⊻0. (5.6) Hence, [𝑥]_𝑆=[0]_𝑆.

Proposition 5.6. Let 𝑆 be a ⊻-closed subset in the bounded Hilbert algebra 𝐴. One denotes (𝑆]={𝑥∈𝐴∣∃𝑠∈𝑆,𝑥≤𝑠} the least decreasing subset generated by 𝑆. Then, (𝑆] is a ⊻-closed subset and it is the complement in 𝐴 of the union of all maximal systems which do not intersect 𝑆.

Proof. It is obvious that (𝑆] is a ⊻-closed subset. We denote the complement in 𝐴 of the union of all maximal systems which do not intersect 𝑆 by 𝑆^′. We shall prove that (𝑆]=𝑆^′. Let 𝑥∈(𝑆]. Then, 𝑥≤𝑠 for some 𝑠∈𝑆. If we suppose that 𝑥∉𝑆^′, there exists a maximal system 𝑀 which does not intersect 𝑆 and 𝑥∈𝑀. But then, 𝑠∈𝑀, false. Conversely, if 𝑥∈𝑆^′, then 𝑥∉𝑀 for each maximal system 𝑀 with 𝑀∩𝑆=∅. If [𝑥)∩𝑆=∅, from Proposition 3.7, there exists a maximal deductive system 𝑀₁ such that 𝑥∈𝑀₁ and 𝑀₁ does not intersect 𝑆, false. Thus, [𝑥) intersects 𝑆 which means that 𝑥∈(𝑆].

In [6], 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 𝐴 be a bounded Hilbert algebra and 𝑆 a ⊻-closed subset in 𝐴. Then, for each bounded Hilbert algebra 𝐵 and for any morphism of bounded Hilbert algebras 𝑓∶𝐴→𝐵 with 𝑓(𝑠)=0, for all 𝑠∈𝑆, there exists an unique morphism 𝑔∶𝐴_𝑆→𝐵 such that 𝑔∘𝑝_𝑆=𝑓 which means that the following diagram is commutative:

(5.7)

Proposition 5.8. Let 𝑆 and 𝑇 be two ⊻-closed subsets in the bounded Hilbert algebra 𝐴, 𝑆⊆𝑇 and we consider the morphism of Hilbert algebras 𝑓∶𝐴_𝑆→𝐴_𝑇 defined by 𝑓([𝑥]_𝑆)=[𝑥]_𝑇,  for all 𝑥∈𝐴. The following statements are equivalent:(i)𝑓 is bijective,(ii)𝑇⊆(𝑆],(iii)for each maximal deductive system 𝑀 with 𝑀∩𝑇≠∅, 𝑀∩𝑆≠∅.

Proof. (i)→(ii) Let 𝑡∈𝑇. Since 𝑓([𝑡]_𝑆)=[𝑡]_𝑇=[0]_𝑇 and 𝑓 is bijective, [𝑡]_𝑆=[0]_𝑆. Hence, from Lemma 5.5, 𝑡∈(𝑆].
(ii)→(iii) Let 𝑀 be a maximal system with 𝑀∩𝑇≠∅. Let 𝑡∈𝑇∩𝑀. Since 𝑇⊆(𝑆], there exists 𝑠∈𝑆 with 𝑡≤𝑠. Hence, 𝑠∈𝑆∩𝑀 and we get 𝑀∩𝑆≠∅.
(iii)→(ii) Let 𝑡∈𝑇. If 𝑡∉(𝑆], then [𝑡)∩𝑆=∅. From Proposition 3.7, there exists a maximal deductive system 𝑀 with [𝑡)⊆𝑀 and 𝑀∩𝑆=∅. Then, since 𝑡∈𝑇∩𝑀, we reach a contradiction.
(ii)→(i) Let [𝑥]_𝑇=[𝑦]_𝑇. Thus, there exists 𝑡∈𝑇 such that 𝑡⊻𝑥=𝑡⊻𝑦. Since 𝑡≤𝑠 for some 𝑠∈𝑆, we obtain that 𝑠^∗≤𝑡^∗. It results that: (𝑠⊻𝑡)⊻𝑥=𝑠⊻(𝑡⊻𝑥)=𝑠^∗→(𝑡^∗→𝑥)=(𝑠^∗→𝑡^∗)→(𝑠^∗→𝑥)=1→(𝑠⊻𝑥)=𝑠⊻𝑥. (5.8) Then, 𝑠⊻𝑥=(𝑠⊻𝑡)⊻𝑥=(𝑠⊻𝑡)⊻𝑦=𝑠⊻𝑦. Hence, [𝑥]_𝑆=[𝑦]_𝑆 and 𝑓 is bijective.

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

Proof. Let 𝑀={𝑥∈𝐴∣[𝑥]_𝑆∈𝒟(𝐴_𝑆)}. Because 𝑀 is the inverse image of the deductive system 𝒟(𝐴_𝑆) by means of the canonical map, it is a deductive system in 𝐴. In our case, 𝒟(𝐴_𝑆) is maximal in 𝐴_𝑆. Hence, 𝑀 is also maximal.
Let us consider now (𝑆], the least decreasing ⊻-closed subset generated by 𝑆. We prove that (𝑆]=𝐴⧵𝑀. Let 𝑥∈𝐴⧵𝑀. Then, 𝑥∉𝑀 if and only if [𝑥]_𝑆∉𝒟(𝐴_𝑆) which is equivalent to [𝑥]_𝑆=[0]_𝑆, as Lemma 3.2 states. Then, from Lemma 5.5, 𝑥∈(𝑆]. Finally, from Proposition 5.8, 𝐴_𝑆≃𝐴_𝑀.

Proposition 5.4 and Proposition 5.9 lead to the following result.

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

Now, we return to the previous subsets 𝑟(𝑥) in Max(𝐴) and 𝑠(𝑥) in Ir(𝐴) where 𝐴 is a bounded Hilbert algebra. For each 𝑥∈𝐴, one defines the sets: 𝑆_𝑥=𝐴⧵_{𝑀∈𝑟(𝑥)}𝑀,𝑇_𝑥=𝐴⧵_{𝐼∈𝑠(𝑥)}𝐼. (5.9)

Lemma 5.11. For each 𝑥∈𝐴, the following statements hold:(1)𝑆_𝑥 are decreasing ⊻-closed subsets. In fact, 𝑆_𝑥={𝑎∈𝐴∣𝑎≤𝑥^∗}.(2)𝑇_𝑥=𝑆_𝑥.

Proof. (1) Let 𝑎,𝑏∈𝑆_𝑥. Then, 𝑎,𝑏∉𝑀, for each 𝑀∈𝑟(𝑥). Since 𝑀 is maximal, 𝑎⊻𝑏∉𝑀, for each 𝑀 and so, 𝑎⊻𝑏∈𝑆_𝑥. Let now consider 𝑎≤𝑏 with 𝑏∈𝑆_𝑥. If 𝑎∉𝑆_𝑥, there is a maximal deductive system 𝑀∈𝑟(𝑥) such that 𝑎∈𝑀. Then, 𝑏∈𝑀 and so, 𝑏∉𝑆_𝑥, false.
Since 𝑟(𝑥^∗∗)=𝑟(𝑥), we have that 𝑆_𝑥∗∗=𝑆_𝑥. As 𝑥^∗∈𝑆_𝑥 and 𝑆_𝑥 is a decreasing ⊻-closed set, each 𝑎≤𝑥^∗ is an element of 𝑆_𝑥. Next, let's consider that 𝑎≤𝑥^∗ and there exists 𝑀∈𝑟(𝑥) such that 𝑎∈𝑀. Then, 𝑥^∗∈𝑀 and, since 𝑥∈𝑀, we get 𝑀=𝐴, false.
(2) 𝑇_𝑥⊆𝑆_𝑥, since each maximal deductive system is an irreducible one. Let 𝑎≤𝑥^∗. If we suppose that 𝑎∉𝑇_𝑥, there is an irreducible deductive system 𝐼∈𝑠(𝑥) with 𝑎∈𝐼. Then, 𝑥^∗∈𝐼 and, finally, 𝐼=𝐴, false.

For each 𝑥₁,…,𝑥_𝑛∈𝐴, we define also the sets 𝑇_{𝑥1,…,𝑥𝑛} as the complement in 𝐴 of the union of all irreducible deductive systems 𝐼∈⋂^𝑛_𝑖=1𝑠(𝑥_𝑖).

Lemma 5.12. With the previous notations, 𝑇_{𝑥1,…,𝑥𝑛}=𝑇_{⋀𝑖=1,𝑛𝑥𝑖}. (5.10)

Proof. Let 𝑎∈𝑇_{𝑥1,…,𝑥𝑛}. Then, 𝑎∉𝐼 for all 𝐼∈𝑠(𝑥_𝑖) with 𝑖=1,…,𝑛. Presuming that 𝑎∉𝑇_⋀^𝑛_{𝑖=1𝑥𝑖}, from Lemma 5.11, 𝑎≰0𝑥00092(⋀^𝑛_𝑖=1𝑥_𝑖)^∗. Hence, there is an irreducible deductive system 𝐽 with 𝑎∈𝐽 and (⋀^𝑛_𝑖=1𝑥_𝑖)^∗∉𝐽, as Theorem 2.12 states. Since 𝐽∨[⋀^𝑛_𝑖=1𝑥_𝑖) is a proper deductive system, it is included into a maximal one. Let 𝑀 be such a maximal deductive system.
From Theorem 2.4, ⋀^𝑛_𝑖=1𝑥_𝑖≤𝑥_𝑖^∗∗ for all 1≤𝑖≤𝑛. From ⋀^𝑛_𝑖=1𝑥_𝑖∈𝑀 we obtain that 𝑥_𝑖^∗∗∈𝑀, hence 𝑥_𝑖∈𝑀, for all 𝑖. Then 𝑀∈⋂^𝑛_𝑖=1𝑠(𝑥_𝑖), and since 𝑎∈𝐽⊆𝑀 we contradict the assumption that 𝑎∈𝑇_{𝑥1,…,𝑥𝑛}.
For the converse inclusion, let us presume that 𝑎∈𝑇_⋀^𝑛_{𝑖=1𝑥𝑖} and 𝑎∉𝑇_{𝑥1,…,𝑥𝑛}. Then, 𝑎≤(⋀^𝑛_𝑖=1𝑥_𝑖)^∗ and there exists 𝐽∈⋂^𝑛_𝑖=1𝑠(𝑥_𝑖) with 𝑎∈𝐽. From Proposition 3.4, since each 𝑥_𝑖∈𝐽,⋀^𝑛_𝑖=1𝑥_𝑖∈𝐽. Hence, (⋀^𝑛_𝑖=1𝑥_𝑖)^∗∉𝐽 and then 𝑎∉𝐽. This contradiction ends the proof.

On the base of the topological space Max(𝐴) we can define a presheaf which means a contravariant functor from Max(𝐴) in the category of Hilbert algebras. We consider the open subsets 𝑟(𝑥)⊆𝑟(𝑦). Then, 𝑆_𝑦⊆𝑆_𝑥.

For 𝑥∈𝐴, let 𝐴_𝑥 be the Hilbert algebra of fractions with respect to the ⊻-closed subset 𝑆_𝑥.

Taking into account the result obtained in Proposition 5.7, we can consider the following commutative diagram:

(5.11) where 𝑝_𝑥, 𝑝_𝑦 are the canonical surjections and 𝜑_𝑦,𝑥∶𝐴_𝑦→𝐴_𝑥 is defined by 𝜑_𝑦,𝑥([𝑎]_𝑆𝑦)=[𝑎]_𝑆𝑥, for all 𝑎∈𝐴.

Hence, each 𝑟(𝑥) is put into correspondence with 𝐴_𝑥 and the morphism 𝜑_𝑦,𝑥∶𝐴_𝑦→𝐴_𝑥 commutes the following diagram:

(5.12)

Lemma 5.13. For 𝑀∈Max(𝐴), the set {𝑆_𝑥∣𝑥∈𝑀} is directed.

Proof. For 𝑥,𝑦∈𝑀, using Proposition 3.4, 𝑥⊼𝑦∈𝑀. Hence, 𝑆_𝑥,𝑆_𝑦⊆𝑆_𝑥⊼𝑦 since 𝑟(𝑥⊼𝑦)=𝑟(𝑥)∩𝑟(𝑦).

We define now the ⊻-closed subset 𝑆_𝑀=⋃_𝑥∈𝑀𝑆_𝑥, for each maximal deductive system 𝑀. Since 𝑆_𝑥⊆𝑆_𝑀, for each 𝑥∈𝑀, there exists a morphism 𝜃_𝑥∶𝐴_𝑥→𝐴_𝑆𝑀 with 𝜃_𝑥∘𝑝_𝑥=𝑝_𝑆𝑀 as Proposition 5.7 states. It is obvious that each morphism 𝜃_𝑥 is surjective.

Let us consider the family of morphisms (𝜃_𝑥∶𝐴_𝑥→𝐴_𝑆𝑀)_𝑥∈𝑀. For each 𝑥,𝑦∈𝑀 such that 𝑆_𝑦⊆𝑆_𝑥, we get (𝜃_𝑥∘𝜑_𝑦,𝑥)([𝑎]_𝑆𝑦)=𝜃_𝑥([𝑎]_𝑆𝑥)=(𝜃_𝑥∘𝑝_𝑥)(𝑎)=[𝑎]_𝑆𝑀=𝜃_𝑦([𝑎]_𝑆𝑦). (5.13) Hence, 𝜃_𝑥∘𝜑_𝑦,𝑥=𝜃_𝑦, for all 𝑥,𝑦∈𝑀 with 𝑆_𝑦⊆𝑆_𝑥.

Let (𝑖_𝑥∶𝐴_𝑥→lim_{→𝑥∈𝑀}𝐴_𝑥)_𝑥∈𝑀 be the direct limit of the directed system (𝐴_𝑥,𝜑_𝑦,𝑥). Since it verifies also a property of universality, there exists a unique morphism 𝜃∶lim_{→𝑥∈𝑀}𝐴_𝑥→𝐴_𝑆𝑀 which commutes the diagram:

(5.14)

Theorem 5.14. With the previous notations, 𝐴_𝑀≃lim_{→𝑥∈𝑀}𝐴_𝑥.

Proof. Let 𝑠∈𝑆_𝑀. Then, there exists 𝑥₀∈𝑀 with 𝑠∈𝑆_𝑥0 which means that 𝑠∉⋃_{𝐽∈𝑟(𝑥0)}𝐽. Hence, 𝑠∉𝐽, for all 𝐽∈𝑟(𝑥₀). We define Ψ∶𝐴→lim_{→𝑥∈𝑀}𝐴_𝑥 by Ψ=𝑖_𝑥0∘𝑝_𝑥0.
We prove that the definition of Ψ is not depending on the choosing of 𝑥₀∈𝑀. To realize that, let 𝑦₀∈𝑀 such that 𝑠∈𝑆_𝑦0. We have to verify that 𝑖_𝑥0∘𝑝_𝑥0=𝑖_𝑦0∘𝑝_𝑦0. Since 𝑥₀,𝑦₀∈𝑀, as in Lemma 5.13, 𝑧₀=𝑥₀⊼𝑦₀∈𝑀 and then 𝑆_𝑥0, 𝑆_𝑦0⊆𝑆_𝑧0.
The following diagram is a commutative one: (5.15)
Indeed, 𝑖_𝑥0∘𝑝_𝑥0=(𝑖_𝑧0∘𝜑_𝑥0,𝑧0)∘𝑝_𝑥0=𝑖_𝑧0∘𝑝_𝑧0=𝑖_𝑧0∘(𝜑_𝑦0,𝑧0∘𝑝_𝑦0)=𝑖_𝑦0∘𝑝_𝑦0 because the family of morphisms (𝑝_𝑥∶𝐴→𝐴_𝑥)_𝑥∈𝑀 verifies the relation 𝑝_𝑥=𝜑_𝑦,𝑥∘𝑝_𝑦, for each pairs (𝑥,𝑦) such that 𝑆_𝑦⊆𝑆_𝑥. In conclusion, Ψ is well defined.
Let 𝑥₀∈𝑀 with 𝑠∈𝑆_𝑥0. Then Ψ(𝑠)=𝑖_𝑥0(𝑝_𝑥0(𝑠))=𝑖_𝑥0([0]_𝑆𝑥0)=0 where 0 is the equivalence class of 0 in the inductive limit. From the property of universality of algebras of fractions, there exists an unique morphism 𝜃^′∶𝐴_𝑆𝑀→lim_{→𝑥∈𝑀}𝐴_𝑥 such that 𝜃^′([𝑎]_𝑆𝑀)=Ψ(𝑎).
From 𝜃^′∘𝑝_𝑆𝑀=Ψ and 𝜃∘Ψ=𝑝_𝑆𝑀, we get that 𝜃∘𝜃^′=1_𝐴𝑆𝑀. The relation (𝜃^′∘𝜃)∘Ψ=𝜃^′∘𝑝_𝑆𝑀=Ψ implies that (𝜃^′∘𝜃)∘𝑖_𝑥=𝑖_𝑥, for each 𝑥∈𝑀. Since 𝑖_𝑥∘𝜑_𝑦,𝑥=𝑖_𝑦 for all 𝑥,𝑦∈𝑀 such that 𝑆_𝑦⊆𝑆_𝑥, we have obtained two morphisms 𝜃^′∘𝜃 and 1_{lim→𝑥∈𝑀𝐴𝑥} which commute the diagram: (5.16)
Using the property of universality of the inductive limit of the directed system (𝐴_𝑥,𝜑_𝑦,𝑥), we get that 𝜃^′∘𝜃=1_{lim→𝑥∈𝑀𝐴𝑥}. Hence, 𝜃^′ is the inverse of the morphism 𝜃 and 𝐴_𝑆𝑀≃lim_{→𝑥∈𝑀}𝐴_𝑥.
To end the proof, we show that 𝑆_𝑀=𝐴⧵𝑀. Let 𝑦∈𝑆_𝑀. Then, 𝑦∉𝐾, where 𝐾∈𝑟(𝑥^′) with 𝑥^′∈𝑀. Hence, 𝑦∉𝑀.
Conversely, since 𝑀 is maximal, for 𝑦∉𝑀 we have 𝑦^∗∈𝑀. From Lemma 5.11 we obtain 𝑦∈𝑆_𝑦∗⊆𝑆_𝑀.

We now repeat the previous construction for irreducible deductive systems. From Lemmas 5.11 and 5.12 we know that for each 𝑛≥1 and each 𝑥₁,…,𝑥_𝑛∈𝐴, 𝑇_{𝑥1,…,𝑥𝑛} is a decreasing ⊻-closed subset in 𝐴.

Lemma 5.15. For 𝐼∈Ir(𝐴), the class {𝑇_{𝑥1,…,𝑥𝑛}∣⋂^𝑛_𝑖=1𝑠(𝑥_𝑖)∈𝑉_𝐼} is directed.

Proof. If we consider 𝑇_⋀^𝑛_{𝑖=1𝑥𝑖} and 𝑇_⋀^𝑚_{𝑗=1𝑦𝑗} be two elements of this class, Lemmas 5.11 and 5.13 imply that 𝑧=⋀^𝑛,𝑚_𝑖,𝑗=1(𝑥_𝑖⊼𝑦_𝑗)∈𝐼 and 𝑇_⋀^𝑛_{𝑖=1𝑥𝑖}, 𝑇_⋀^𝑚_{𝑗=1𝑦𝑗}⊆𝑇_𝑧.

Now, let 𝐼 be an irreducible deductive system in 𝐴. We denote the union of all the ⊻-subsets 𝑇_{𝑥1,…,𝑥𝑛}, when ⋂^𝑛_𝑖=1𝑠(𝑥_𝑖)∈𝑉_𝐼 and 𝑛≥1 with 𝑇_𝐼. Then, 𝑇_𝐼 is a decreasing ⊻-subset in 𝐴.

Theorem 5.16. With the previous notations, for an irreducible deductive system 𝐼 in the bounded Hilbert algebra 𝐴,𝐴_̃𝐼≃𝐴_𝑇𝐼≃lim_{→⋂𝑛𝑖=1𝑠(𝑥𝑖)∈𝑉𝐼}𝑇_⋀^𝑛_{𝑖=1𝑥𝑖}. (5.17)

Proof. From Proposition 3.12, 𝐼̃ is a decreasing ⊻-closed subset.
Since 𝑇_𝐼 is a decreasing ⊻-closed subset in 𝐴, we obtain, as in the proof of Theorem 5.14, that 𝐴_𝑇𝐼≃lim_{→⋂𝑛𝑖=1𝑠(𝑥𝑖)∈𝑉𝐼}𝑇_⋀^𝑛_{𝑖=1𝑥𝑖}. (5.18) Finally, we prove that 𝑇_𝐼=𝐼.̃
For 𝑥∈𝐼̃, 𝑥^∗∈𝐼. Then, 𝑥∈𝑇_𝑥∗={𝑎∈𝐴,𝑎≤𝑥^∗∗}. Since 𝑇_𝑥∗⊆𝑇_𝐼, we obtain that 𝑥∈𝑇_𝐼. Conversely, let 𝑥∈𝑇_𝐼. This implies that there exists an element ⋀^𝑘_𝑖=1𝑦_𝑖∈𝐼 such that 𝑥∈𝑇_⋀^𝑘_{𝑖=1𝑦𝑖}. Hence, 𝑥≤(⋀^𝑘_𝑖=1𝑦_𝑖)^∗. But, ⋀^𝑘_𝑖=1𝑦_𝑖=(⋀^𝑘_𝑖=1𝑦_𝑖)^∗∗≤𝑥^∗. As ⋀^𝑘_𝑖=1𝑦_𝑖∈𝐼, we get 𝑥^∗∈𝐼 which is equivalent to 𝑥∈𝐼.̃ Thus, 𝑇_𝐼=𝐼.̃