# Hilbert Algebras of Fractions

Authors: Christina-Theresia Dan
Journal: International Journal of Mathematics and Mathematical Sciences
 (2009)

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## Abstract

LetA be a bounded Hilbert algebra and S
a ∨¯-closed subset of A. The Hilbert algebra of fractions AS 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.


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## Body

## 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.
LetA 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 algebraA 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 algebraA has a smallest element, denoted by 0, it is called a bounded Hilbert algebra.Definition 2.2.
IfA1,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 algebraA 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)
ifx≤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 algebraA, 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 subsetD 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 algebraA, 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]).
LetD 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 inA with Ds(A).IfX 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]).
LetA 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,…,an∈S such that a1→(a2→(⋯(an→x)⋯))=1.In particular, if we consider the setS={a}, the deductive system
(2.1)[a)={x∈A∣a≤x}
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]).
IfA is a Hilbert algebra, D∈Ds(A), and a∈A, then
(2.2)[a)∨D={x∈A∣a→x∈D}.In [10] is defined the concept of ideals in a Hilbert algebra as follows.Definition 2.9 (see [10]).
A nonempty subsetI 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]).
LetA 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 inA.For a Hilbert algebraA 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]).
LetA be a Hilbert algebra.(i)
IfD 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)
Ifa≰b, there exists an irreducible deductive system I such that a∈I and b∈A∖I.(iii)
IfA is bounded, each proper deductive system is included in a maximal one.Lemma 2.13 (see [3]).
LetA be a bounded Hilbert algebra. The following statements are equivalent:(i)
M is a maximal deductive system in A.(ii)
Forx,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 system𝒟(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 belocal 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 algebraA is local if and only if for x,y∈A, x⊻y=1 implies x**=1 or y**=1.Proof.
IfA 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.
LetA be a local bounded Hilbert algebra with 𝒟(A) its single maximal deductive system. Then, 𝒟(A)=A∖{0}.Proof.
Letx∈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 systemD of a Hilbert algebra is the intersection of the irreducible deductive systems which contain D.Proposition 3.4.
IfD 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 existx,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]).
LetS 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.
LetD 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 considerD 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.
LetA 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(3.1)T={F∣Fdeductive  system,D⊆F,F∩S=∅}.
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 relationy→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.
Sincex→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.
LetS 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 thatx⊻y∈S if and only if x,y∈S for a decreasing ⊻-closed subset S.For an arbitrary elementx 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 D̃ as follows:
(3.2)D̃={x∈A∣D∨[x**)=A}.
Then, x∈D̃ 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∉D̃, otherwise D=A.Lemma 3.10.
LetD be a deductive system in a bounded Hilbert algebra A. Then, x∈D̃ if and only if x*∈D.Proof.
Ifx∈D̃ 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∈D̃.Remark 3.11.
IfD and D̃ intersect, 0∈D and then A=D. Hence, if we consider D to be a proper deductive system, D∩D̃=∅. Thus, D̃⊆A∖D. The equality holds if and only if D is a maximal deductive system. Indeed, D̃=A∖D if and only if A∖D⊆D̃. Hence, for all x∉D,x∈D̃ 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.
D̃ defined as above is a decreasing ⊻-closed subset if D is a proper deductive system in A.Proof.
Firstly, because1∈D, we see that 0∈D̃. Let x≤y and y∈D̃. Since y*≤x* and y*∈D, we get x*∈D. Thus, x∈D̃. Now we consider x,y∈D̃. Then, x*,y*∈D and, from Proposition 3.4, x*⌅y*∈D.
But, from Theorem2.4 (13), x*⌅y*=(x⊻y)*∈D which proves that x⊻y∈D̃.For a proper deductive systemD we define
(3.3)D0={x∈A∣x**∈D}.
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 considerD a proper deductive system. Using the following chain of equivalences
(3.4)x∈D̃⇔x*∈D⇔x*∈D0⇔x∈D̃0,
we obtain that D̃=D̃0.Proposition 3.14.
For a proper deductive systemD in A there is an irreducible deductive system J such that D⊆J⊆A∖D̃.Proof.
If we considerx∈D0, then x**∈D. Hence, x*∉D, else 0∈D and D=A. Then x∉D̃ and so D⊆D0⊆A∖D̃. Thus, F, the family of all deductive systems which contain D and are included in A∖D̃, 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⊈A∖D̃. Let a∈J1,b∈J2 with a,b∉A∖D̃. From Proposition 3.12, D̃ is ⊻-closed. Hence, since a,b∈D̃ we obtain a⊻b∈D̃. But a⊻b∈J1∩J2=J and this contradicts the fact that J∈F.


## 4. The SpacesMax(A) and Ir(A)

LetA be a bounded Hilbert algebra. For each x∈A, we define the set:
(4.1)r(x)={M∈Max(A)∣x∈M}.We will considerMax(A) as a topological space with the class {r(x)∣x∈A} as a base.Theorem 4.1 (see [3]).
For eachx,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.
Forx,y∈A, r(x)=r(y) if and only if x**=y**.Proof.
Letr(x)=r(y). It means that for an arbitrary maximal deductive system M, x∈M if and only if y∈M. Assuming that x**≰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 obtainx**≤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 setB 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 Bx={B∣B∈B,x∈B} is a fundamental system of neighborhoods for x.In our case, we have considered onMax(A) the topology generated by the basis {r(x)∣x∈A}. Using this property, the following lemma results.Lemma 4.3.
LetM∈Max(A). Then, VM={r(x)∣x∈M} is a fundamental system of neighborhoods for M.Theorem 4.4 (see [9]).
Every Hilbert algebraA 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 Lemma4.3.Lemma 4.5.
LetI be an irreducible deductive system in the bounded Hilbert algebra A. Then,
(4.2)VI={⋂i=1ns(xi)∣xi∈I,1≤i≤n,n≥1}
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.LetA be a bounded Hilbert algebra and S be a ⊻-closed subset in A. A congruence relation θS on A is defined by (x,y)∈θS⇔∃s∈S, s⊻x=s⊻y. The corresponding quotient Hilbert algebra A/θS is denoted by AS and it is called the Hilbert algebra of fractions ofAwith respect to the⊻-closed subsetS. The congruence class of x in AS will be denoted by [x]S.IfD is a deductive system in A, it is easy to verify that DS={[x]S∣x∈D} is a deductive system in AS.LetM 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 AM.We know that a deductive systemD induces a congruence relation θD on A defined by: (x,y)∈θD⇔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.
LetA be a Hilbert algebra, D a deductive system and S a ⊻-closed subset in A. On A/D we define the following relation:
(5.1)(x/D,y/D)∈θS′⇔∃s∈S,(s⊻x)/D=(s⊻y)/D.
Then, θS′ is a congruence relation on A/D.Proof.
LetpD:A→A/D be the canonical morphism of Hilbert algebras. We easily see that SD=pD(S)={pD(s)∣s∈S} is a ⊻-closed subset of A/D. Then, (x/D,y/D)∈θSD⇔∃t∈SD 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)∈θS′.
Hence,θS′ is exactly the congruence defined in the Hilbert algebra A/D by means of the ⊻-closed subset SD=pD(S).The quotient Hilbert algebra regarding the congruence relationθS′ will be denoted by (A/D)S and the corresponding congruence classes will be denoted by [x/D]S.For a maximal deductive systemM, let S=A∖M be the related ⊻-closed subset. Let pM:A→AM be the canonical morphism defined by pM(x)=[x]M for each x∈A.Lemma 5.2.
IfD is a deductive system in the bounded Hilbert algebra A, then
(5.2)D=⋂M∈Max(A)pM-1(DM).Proof.
SinceD⊆pM-1(DM) for each maximal deductive system, we obtain one inclusion. To prove the converse inclusion, let us consider x∈pM-1(DM), for each maximal deductive system M. Then, pM(x)∈⋂M∈Max(A)DM. 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}⊈M, 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.
LetD1, 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,
(5.3)D1=⋂M∈Max(A)pM-1((D1)M)⊆⋂M∈Max(A)pM-1((D2)M)=D2.
(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,
(5.4)s1⊻s⊻t∉M,(s1⊻s⊻t)⊻(y→x),(s1⊻s⊻t)⊻(x→y)∈D2.
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 thatA/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.
IfM is a maximal deductive system in the Hilbert algebra A, the Hilbert algebra of fractions AM is local.Proof.
We show that𝒟(AM) 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 fractionsAS, [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,
(5.5)s⊻x=(s⊻x)⊻x=s⊻(x⊻x)=s⊻x**
and then,
(5.6)s⊻x=s⊻x**=(s⊻x)**=s**=s⊻0.
Hence, [x]S=[0]S.Proposition 5.6.
LetS 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.
LetA 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:AS→B such that g∘pS=f which means that the following diagram is commutative:(5.7)Proposition 5.8.
LetS and T be two ⊻-closed subsets in the bounded Hilbert algebra A, S⊆T and we consider the morphism of Hilbert algebras f:AS→AT 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 systemM 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:
(5.8)(s⊻t)⊻x=s⊻(t⊻x)=s*→(t*→x)=(s*→t*)→(s*→x)=1→(s⊻x)=s⊻x.
Then, s⊻x=(s⊻t)⊻x=(s⊻t)⊻y=s⊻y. Hence, [x]S=[y]S and f is bijective.Proposition 5.9.
LetA be a bounded Hilbert algebra and S a ⊻-closed subset in A. If AS is local, there exists a maximal system M such as AS and AM are isomorphic.Proof.
LetM={x∈A∣[x]S∈𝒟(AS)}. Because M is the inverse image of the deductive system 𝒟(AS) by means of the canonical map, it is a deductive system in A. In our case, 𝒟(AS) is maximal in AS. 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(AS) 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, AS≃AM.Proposition5.4 and Proposition 5.9 lead to the following result.Theorem 5.10.
A bounded Hilbert algebra of fractionsAS is local if and only if there exists a maximal deductive system M in A such that AS and AM are isomorphic.Now, we return to the previous subsetsr(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:
(5.9)Sx=A∖⋃M∈r(x)M,Tx=A∖⋃I∈s(x)I.Lemma 5.11.
For eachx∈A, the following statements hold:(1)
Sx are decreasing ⊻-closed subsets. In fact, Sx={a∈A∣a≤x*}.(2)
Tx=Sx.Proof.
(1) Let a,b∈Sx. Then, a,b∉M, for each M∈r(x). Since M is maximal, a⊻b∉M, for each M and so, a⊻b∈Sx. Let now consider a≤b with b∈Sx. If a∉Sx, there is a maximal deductive system M∈r(x) such that a∈M. Then, b∈M and so, b∉Sx, false.
Sincer(x**)=r(x), we have that Sx**=Sx. As x*∈Sx and Sx is a decreasing ⊻-closed set, each a≤x* is an element of Sx. 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) Tx⊆Sx, since each maximal deductive system is an irreducible one. Let a≤x*. If we suppose that a∉Tx, there is an irreducible deductive system I∈s(x) with a∈I. Then, x*∈I and, finally, I=A, false.For eachx1,…,xn∈A, we define also the sets Tx1,…,xn as the complement in A of the union of all irreducible deductive systems I∈⋂i=1ns(xi).Lemma 5.12.
With the previous notations,(5.10)Tx1,…,xn=T⋀̅i=1,nxi.Proof.
Leta∈Tx1,…,xn. Then, a∉I for all I∈s(xi) with i=1,…,n. Presuming that a∉T⋀̅i=1nxi, from Lemma 5.11, a≰(⋀̅i=1nxi)*. Hence, there is an irreducible deductive system J with a∈J and (⋀̅i=1nxi)*∉J, as Theorem 2.12 states. Since J∨[⋀̅i=1nxi) is a proper deductive system, it is included into a maximal one. Let M be such a maximal deductive system.
From Theorem2.4, ⋀̅i=1nxi≤xi** for all 1≤i≤n. From ⋀̅i=1nxi∈M we obtain that xi**∈M, hence xi∈M, for all i. Then M∈⋂i=1ns(xi), and since a∈J⊆M we contradict the assumption that a∈Tx1,…,xn.
For the converse inclusion, let us presume thata∈T⋀̅i=1nxi and a∉Tx1,…,xn. Then, a≤(⋀̅i=1nxi)* and there exists J∈⋂i=1ns(xi) with a∈J. From Proposition 3.4, since each xi∈J,⋀̅i=1nxi∈J. Hence, (⋀̅i=1nxi)*∉J and then a∉J. This contradiction ends the proof.On the base of the topological spaceMax(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, Sy⊆Sx.Forx∈A, let Ax be the Hilbert algebra of fractions with respect to the ⊻-closed subset Sx.Taking into account the result obtained in Proposition5.7, we can consider the following commutative diagram:(5.11)
where px, py are the canonical surjections and φy,x:Ay→Ax is defined by φy,x([a]Sy)=[a]Sx, for all a∈A.Hence, eachr(x) is put into correspondence with Ax and the morphism φy,x:Ay→Ax commutes the following diagram:(5.12)Lemma 5.13.
ForM∈Max(A), the set {Sx∣x∈M} is directed.Proof.
Forx,y∈M, using Proposition 3.4, x⌅y∈M. Hence, Sx,Sy⊆Sx⌅y since r(x⌅y)=r(x)∩r(y).We define now the⊻-closed subset SM=⋃x∈MSx, for each maximal deductive system M. Since Sx⊆SM, for each x∈M, there exists a morphism θx:Ax→ASM with θx∘px=pSM as Proposition 5.7 states. It is obvious that each morphism θx is surjective.Let us consider the family of morphisms(θx:Ax→ASM)x∈M. For each x,y∈M such that Sy⊆Sx, we get
(5.13)(θx∘φy,x)([a]Sy)=θx([a]Sx)=(θx∘px)(a)=[a]SM=θy([a]Sy).
Hence, θx∘φy,x=θy, for all x,y∈M with Sy⊆Sx.Let(ix:Ax→lim→x∈MAx)x∈M be the direct limit of the directed system (Ax,φy,x). Since it verifies also a property of universality, there exists a unique morphism θ:lim→x∈MAx→ASM which commutes the diagram:(5.14)Theorem 5.14.
With the previous notations,AM≃lim→x∈MAx.Proof.
Lets∈SM. Then, there exists x0∈M with s∈Sx0 which means that s∉⋃J∈r(x0)J. Hence, s∉J, for all J∈r(x0). We define Ψ:A→lim→x∈MAx by Ψ=ix0∘px0.
We prove that the definition ofΨ is not depending on the choosing of x0∈M. To realize that, let y0∈M such that s∈Sy0. We have to verify that ix0∘px0=iy0∘py0. Since x0,y0∈M, as in Lemma 5.13, z0=x0⌅y0∈M and then Sx0, Sy0⊆Sz0.
The following diagram is a commutative one:(5.15)
Indeed,ix0∘px0=(iz0∘φx0,z0)∘px0=iz0∘pz0=iz0∘(φy0,z0∘py0)=iy0∘py0 because the family of morphisms (px:A→Ax)x∈M verifies the relation px=φy,x∘py, for each pairs (x,y) such that Sy⊆Sx. In conclusion, Ψ is well defined.
Letx0∈M with s∈Sx0. Then Ψ(s)=ix0(px0(s))=ix0([0]Sx0)=0̃ where 0̃ 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′:ASM→lim→x∈MAx such that θ′([a]SM)=Ψ(a).
Fromθ′∘pSM=Ψ and θ∘Ψ=pSM, we get that θ∘θ′=1ASM. The relation (θ′∘θ)∘Ψ=θ′∘pSM=Ψ implies that (θ′∘θ)∘ix=ix, for each x∈M. Since ix∘φy,x=iy for all x,y∈M such that Sy⊆Sx, we have obtained two morphisms θ′∘θ and 1lim→x∈MAx which commute the diagram:
(5.16)
Using the property of universality of the inductive limit of the directed system(Ax,φy,x), we get that θ′∘θ=1lim→x∈MAx. Hence, θ′ is the inverse of the morphism θ and ASM≃lim→x∈MAx.
To end the proof, we show thatSM=A∖M. Let y∈SM. Then, y∉K, where K∈r(x′) with x′∈M. Hence, y∉M.
Conversely, sinceM is maximal, for y∉M we have y*∈M. From Lemma 5.11 we obtain y∈Sy*⊆SM.We now repeat the previous construction for irreducible deductive systems. From Lemmas5.11 and 5.12 we know that for each n≥1 and each x1,…,xn∈A, Tx1,…,xn is a decreasing ⊻-closed subset in A.Lemma 5.15.
ForI∈Ir(A), the class {Tx1,…,xn∣⋂i=1ns(xi)∈VI} is directed.Proof.
If we considerT⋀̅i=1nxi and T⋀̅j=1myj be two elements of this class, Lemmas 5.11 and 5.13 imply that z=⋀̅i,j=1n,m(xi⌅yj)∈I and T⋀̅i=1nxi, T⋀̅j=1myj⊆Tz.Now, letI be an irreducible deductive system in A. We denote the union of all the ⊻-subsets Tx1,…,xn, when ⋂i=1ns(xi)∈VI and n≥1 with TI. Then, TI is a decreasing ⊻-subset in A.Theorem 5.16.
With the previous notations, for an irreducible deductive systemI in the bounded Hilbert algebra A,(5.17)AĨ≃ATI≃lim→⋂i=1ns(xi)∈VIT⋀̅i=1nxi.Proof.
From Proposition3.12, Ĩ is a decreasing ⊻-closed subset.
SinceTI is a decreasing ⊻-closed subset in A, we obtain, as in the proof of Theorem 5.14, that
(5.18)ATI≃lim→⋂i=1ns(xi)∈VIT⋀̅i=1nxi.
Finally, we prove that TI=Ĩ.
Forx∈Ĩ, x*∈I. Then, x∈Tx*={a∈A,a≤x**}. Since Tx*⊆TI, we obtain that x∈TI. Conversely, let x∈TI. This implies that there exists an element ⋀̅i=1kyi∈I such that x∈T⋀̅i=1kyi. Hence, x≤(⋀̅i=1kyi)*. But, ⋀̅i=1kyi=(⋀̅i=1kyi)**≤x*. As ⋀̅i=1kyi∈I, we get x*∈I which is equivalent to x∈Ĩ. Thus, TI=Ĩ.



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*Source: 589830-2009-12-24.xml*