THE LOCALIZATION OF COMMUTATIVE (UNBOUNDED) HILBERT ALGEBRAS

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(UNBOUNDED) HILBERT ALGEBRAS

DANA PICIU and C ˘AT ˘ALIN BUS¸NEAG

In this paper we develop a theory of localization for unbounded commutative Hilbert algebras. For bounded case see [4] and [6].

AMS 2000 Subject Classification: 03G25, 18A15, 18C05.

Key words: Hilbert algebra, commutative Hilbert algebra, algebra of fractions, maximal algebra of quotients,∨-closed system, topology, localization algebra.

1. INTRODUCTION

Hilbert algebras are important tools for certain investigations in algebraic logic since they can be considered as fragments of any propositional logic con- taining a logical connective implication and the constant 1 which is considered as the logical value “true”. The concept of Hilbert algebras was introduced in the 50-ties by L. Henkin and T. Skolem (under the name implicative models) for investigations in intuitionistic logics and other non-classical logics. Diego [9] proved that Hilbert algebras form a variety which is locally finite. They were studied from various points of view.

In this paper we develop a theory of localization for commutative (unbounded) Hilbert algebras, and then we deal with generalizations of results which are obtained in the papers [4] and [5] for bounded case.

This paper is organized as follows: In Section 2 we recall the basic defi- nitions and put in evidence many rules of calculus in commutative Hilbert algebras which we need in the rest of paper (especiallyc11–c14). In Section 3 we introduce the commutative Hilbert algebra of fractions relative to a ∨-closed system. In Section 4 we develop a theory for multipliers on a commutative (unbounded) Hilbert algebra. In Section 5 we define the notions of Hilbert algebras of fractions and maximal Hilbert algebra of quotients for a commutative (unbounded) Hilbert algebra. In the least part of this section is proved the existence of the maximal Hilbert algebra of quotients (Theorem 21). In Section 6 we develop a theory of localization for commutative (unbounded) Hilbert algebras. So, for a commutative (unbounded) Hilbert algebra A we

MATH. REPORTS12(62),3 (2010), 285–300

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define the notion of localization Hilbert algebra relative to a topology F on A. In Section 7 we describe the localization Hilbert algebra AF in some special instances.

For a survey relative to localization and fractions in algebra of logic see [15].

2. PRELIMINARIES

In this paper the symbols⇒ and ⇔ will be used for logical implication and logical equivalence, respectively.

Definition 1 ([2]–[6], [9]). A Hilbert algebra is an algebra (A,→,1) of type (2,0) such that the axioms

(a₁) x→(y→x) = 1;

(a2) (x→(y →z))→((x→y)→(x→z)) = 1;

(a3) if x→y=y→x= 1,thenx=y are fulfilled for every x, y, z∈A.

In [9] it is proved that the system of axioms {a₁, a2, a3} is equivalent with the system {a₄, a₅, a₆, a₇}, where

(a₄) x→x= 1;

(a5) 1→x= 1;

(a₆) x→(y→z) = (x→y)→(x→z);

(a7) (x→y)→((y→x)→x) = (y →x)→((x→y)→y).

For examples of Hilbert algebras see [2]–[5] and [7]. If A is a Hilbert algebra, then the relation ≤defined byx ≤y iff x→ y= 1 is a partial order onA(called thenatural order onA); with respect to this order 1 is the largest element of A. A bounded Hilbert algebra is a Hilbert algebra with a smallest element 0; in this case, for x∈Awe denote x^∗=x→0.

Following [2]–[6] and [9], in a Hilbert algebra A we have the following rules of calculus for x, y, z ∈A:

(c₁) x→1 = 1;

(c2) x≤y→x;

(c₃) x≤(x→y)→y;

(c₄) ((x→y)→y)→y =x→y;

(c5) x→y≤(y→z)→(x→z);

(c₆) if x≤y, thenz→x≤z→y and y→z≤x→z;

(c7) x→(y→z) =y→(x→z).

Forx₁, . . . , x_n, x∈A,a convenient technical device is the compact nota- tion x1 →(x2 → (. . .→ (xn→ x). . .) = (x1, . . . , xn;x) which can be defined

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by induction: (x1;x) = x1 → x, (x1, . . . , xn+1;x) = (x1, . . . , xn;xn+1 → x).

Then (see [4]) we have

(c8) ifσis a permutation of{1,2, . . . , n}(n≥2),then (x_σ(1), . . . , x_σ(n);x) = (x1, . . . , xn;x);

(c₉) x→(x₁, . . . , x_n;y) = (x, x₁, . . . , x_n;y).

Definition 2. If A is a Hilbert algebra, a subset D of A is a deductive system ofA if the following axioms are satisfied:

(a8) 1∈D;

(a₉) if x, x→y ∈D, theny∈D.

We denote byDs(A) the set of all deductive systems of A.

For a nonempty subset X ⊆ A, we denote by [X) = T

{D ∈ Ds(A) : X ⊆D}([X) is called thedeductive system generated by X).IfS ={a},with a∈ A,we denote by [a) the deductive system generated by {a} ([a) is called principal).

Proposition1 ([4]). IfA is a Hilbert algebra andX⊆Ais a nonempty subset, then

(c10) [X) ={x∈A:there existx1, . . . , xn∈Xsuch that(x1, . . . , xn;x) = 1}.

In particular, from(c₁₀)we deduce that ifx₁, . . . , x_n∈A,then[{x₁, . . . , x_n}) = {x∈A: (x1, . . . , xn;x) = 1}.If a∈A, then[a) ={x∈A:a≤x}.

Definition 3. IfA₁, A₂ are Hilbert algebras, then f :A₁ → A₂ is called morphism of Hilbert algebras if

(a₁₀) f(x→y) =f(x)→f(y) for everyx, y∈A₁.

Definition 4 ([11], [12]). A Hilbert algebraA is said to be commutative if it satisfies the axiom

(a11) (x→y)→y= (y →x)→x,for everyx, y∈A.

For examples of commutative Hilbert algebras see [11], [12].

Proposition 2 ([11], [12]). If A is a commutative Hilbert algebra, then relative to the natural order, A is a join-semi lattice where, for x, y∈A,

(c₁₁) x∨y = (x→y)→y= (y→x)→x.

Proof. From (c2) and (c3) we have x, y ≤(x → y) → y. Let now t ∈A such that x, y ≤t. From (c6) we deduce t → y ≤ x → y ⇒ (x → y) → y ≤ (t→y)→y= (y →t)→t= 1→t=t, that is, x∨y= (x→y)→y .

Remark 1. If A is a bounded commutative Hilbert algebra, then A is a Boolean algebra (since for every x ∈A from (x→ 0)→0 = (0→x)→x ⇒ x^∗∗=x⇒A is a Boolean algebra, see Lemma 2.7 from [3]).

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Remark 2. IfA1, A2 are commutative Hilbert algebras and f :A1 →A2

is a morphism of Hilbert algebras, then

(c12) f(x∨y) =f(x)∨f(y),for everyx, y∈A1.

Lemma 3. For everyx, y, z ∈A there exists (x→z)∧(y →z) and (c13) (x∨y)→z = (x→z)∧(y→z).

Proof.Sincex, y≤x∨y,by (c₆) we deduce thatx→z, y→z≤(x∨y)→ z. Let now t∈A such that t≤x →z, y →z. Then x, y ≤t→ z ⇒x∨y≤ t→z⇒ t≤(x∨y)→z,that is, (x∨y)→z= (x→z)∧(y→z).

Corollary 4. For every x, y, z∈A we have (c14) x∨(y→z) = (x∨y)→(x∨z).

Proof. By (c₁₃) we have (x∨y) → (x∨z) = (x → (x∨z))∧(y → (x∨z)) = 1∧(y → (x∨z)) = y → (x∨z) = y → ((x → z) → z) and x∨(y→z) = (x→(y→z))→ (y →z) = (y →(x →z))→(y →z) =y → ((x→z)→z),hence (c₁₄) holds.

3. COMMUTATIVE HILBERT ALGEBRA OF FRACTIONS RELATIVE TO A∨-CLOSED SYSTEM

In this section by A we denote a commutative (unbounded) Hilbert algebra.

Definition 5. A nonempty subset S of A will be called ∨-closed system of A if

(a11) x∨y ∈S for everyx, y∈S.

For a ∨-closed system S ⊆A we define the binary relation θS on A by (x, y)∈θ_S iff there is s∈S such thats∨x=s∨y.

Proposition 5. The relationθ_S is a congruence onA.

Proof. Clearly,θS is an equivalence relation onA.To prove the compatibility ofθ_Swith operation→, letx, y, z ∈Asuch that (x, y)∈θ_S(hence there is s∈S such thats∨x=s∨y).By (c₁₄) we deduce s∨(z→x) = (s∨z)→ (s∨x) = (s∨z)→(s∨y) =s∨(z→y),and similarly,s∨(x→z) =s∨(y→z), that is, (z→x, z →y),(x→z, y→z)∈θ_S .

We denote A[S] =A/θS; the commutative Hilbert algebra A[S] will be called Hilbert algebra of fractions of A relative to S. Forx∈Awe denote by [x]θS the equivalence class of x relative to θS. Clearly, in A[S], 1 = [1]θS = {x∈A: (x,1)∈θ_S}={x∈A: there iss∈S such thats∨x= 1}.

Proposition 6. A[S] is a bounded commutative Hilbert algebra, when 0= [s]θS with s∈S.

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Proof.Clearly, ifs, t∈S,sincer=s∨t∈S and r∨s=r∨t⇒[s]θS = [t]_θ_S. To prove that for s∈ S,[s]_θ_S =0,let x ∈A. We have [s]_θ_S ≤[x]_θ_S ⇔ [s]θS ∨[x]θS = [x]θS ⇔ [s∨x]θS = [x]θS which is true because s∨(s∨x) = s∨x.

We denote byp_S:A→A[S] the canonical surjective morphism of Hilbert algebras (defined by ps(x) = [x]θ_S,for everyx∈A).

A[S] verify the following property of universality:

Theorem7. For every bounded commutative Hilbert algebraB and every morphism of Hilbert algebras f :A→B such that f(S) ={0}, there exists a unique morphism of Hilbert algebras f⁰ :A[S]→B such that the diagram:

A −→^p^s A[S]

&

f

.

f⁰

B is commutative (i.e., f⁰◦pS =f).

Proof. Let x, y ∈ A such that [x]θ_S = [y]θ_S. Then there is s ∈ S such that s∨x = s∨y ⇒ f(s∨x) = f(s∨y) ⇒ f(s)∨f(x) = f(s)∨f(y) ⇒ 0∨f(x) = 0∨f(y)⇒f(x) =f(y).

So, f⁰ : A[S] → B defined for x ∈ A by f⁰([x]_θ_S) = f(x) is correctly defined. Clearly, f⁰ is morphism of Hilbert algebras and f⁰ ◦p_s = f. The unicity of f⁰ follows from the surjectivity of ps.

4. MULTIPLIERS ON A COMMUTATIVE HILBERT ALGEBRA

The concept ofmaximal lattice of quotients for a distributive lattice was defined by Schmid in ([16], [17]) taking as a guide-line the construction of complete ring of quotients by partial morphisms introduced by Findlay and Lambek (see [13], p. 36). The central role in the construction of the maximal lattice maximal lattice of quotients for a distributive lattice due to Schmidt ([16] and [17]) is played by the concept of multiplier for a distributive lattice defined by Cornish [7].

In this section we develop a theory for multipliers on a commutative (unbounded) Hilbert algebraA.

Definition 6. A subset S ⊆A is called ∨-subset of A if for everya∈A and x∈S we have a∨x∈S.

We denote byS(A) the set of all∨-subsets ofA.Clearly,Ds(A)⊆S(A) (and more generally, if denote by I(A) the set of all increasing subsets of A (I ∈ I(A) iff x ≤ y and x ∈ I, then y ∈ I), then I(A) ⊆S(A)). Clearly, if D1, D2 ∈S(A),thenD1∩D2∈S(A).

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Lemma 8. If D∈S(A),then (i) 1∈D;

(ii) if x≤y and x∈D,theny ∈D.

Proof. (i) Ifx∈D,since 1∈A,then 1 = 1∨x∈D.

(ii) Ifx≤y,theny =x∨y.

Definition 7. By partial multiplier on A we mean a map f : D → A, where D∈S(A),such that

(a12) f(a∨x) =a∨f(x), for every x∈D anda∈A.

By dom(f) ∈ S(A) we denote the domain of f; if dom(f) = A, f is called total.

To simplify the language, we use multiplier instead partial multiplier using total to indicate that the domain of a certain multiplier isA.

Example1. The maps0,1:A→Adefined by0(x) =xand respectively 1(x) = 1,for everyx∈Aare total multipliers on A.

Example 2. For a∈ A and D ∈S(A), the map fa :D → A defined by f_a(x) =a∨x,for everyx∈Dis a multiplier on A (calledprincipal).

If dom(f_a) =A,we denote f_a by f_a .

Lemma 9. If f :D→A is a multiplier on A (with D∈S(A)),then (i) f(1) = 1;

(ii) for everyx∈D, x≤f(x).

Proof. (i) If in (a₁₂) we put a = 1, we obtain that for every x ∈ D, f(1∨x) = 1∨f(x)⇔f(1) = 1.

(ii) If in (a12) we put a =x, we obtain f(x∨x) = x∨f(x) ⇔ f(x) = x∨f(x)⇔x≤f(x).

ForD ∈S(A), we denoteM(D, A) ={f :D→A :f is a multiplier on A}and M(A) = S

D∈S(A)

M(D, A).

For D₁, D₂ ∈ S(A) and f_i ∈ M(D_i, A), i = 1,2, we define f₁ → f₂ : D1∩D2→A by (f1 →f2)(x) =f1(x)→f2(x),for everyx∈D1∩ D2.

Lemma 10. f1→f2 ∈M(D1∩D2, A).

Proof. If x ∈D₁∩D₂ and a∈A, then (f₁ → f₂)(a∨x) = f₁(a∨x) → f2(a∨x) = (a∨f1(x))→(a∨f2(x))^c=¹⁴ a∨(f₁(x)→f2(x)) =a∨(f₁→f2)(x), that is, f₁ →f₂ ∈M(D₁∩D₂, A).

Corollary 11. (M(A),→,1,0) is a bounded Hilbert algebra.

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Proof. The fact that M(A) is a Hilbert algebra follows immediately from Lemma 10. If D ∈S(A), f ∈M(D, A) and x ∈D, then by Lemma 9, 0(x) =x≤f(x)≤1 =1(x).

Lemma 12. The map v_A :A → M(A) defined by v_A(a) = f_a for every a∈A is a morphism of Hilbert algebras.

Proof. If a, b ∈ A then (fa → fb)(x) = fa(x) → fb(x) = (a∨x) → (b∨x)^c= (a¹⁴ →b)∨x=fa→b(x),so, v_A(a)→v_A(b) =v_A(a→b).

Definition 8. D ⊆ A is called regular if for every x, y ∈ A such that z∨x=z∨y for everyz∈D,then x=y.

For example, A is regular since if x, y ∈ A such that z∨x = z∨y for every z ∈A, then in particular, forz =x we obtain x =x∨y ⇒ y ≤x and analogously for z=y we obtainx≤y,hence x=y.

IfA is bounded and 0∈D,thenD is regular.

We denote R(A) ={D⊆A:Dis a regular subset of A}.

Lemma 13. If D₁, D₂ ∈S(A)∩R(A), thenD₁∩D₂ ∈S(A)∩R(A).

Proof. Let x, y ∈ A such that z∨x = z∨y for every z ∈ D₁ ∩D₂. For every zi ∈ Di, i = 1,2, since z1∨z2 ∈ D1∩D2 we have (z1∨z2)∨x = (z₁∨z₂)∨y⇒z₁∨(z₂∨x) =z₁∨(z₂∨y)⇒z₂∨x=z₂∨y⇒x=y.

Corollary 14. If denote M_r(A) = {f ∈ M(A) : dom(f) ∈ S(A)∩ R(A)}, thenM_r(A) is a Hilbert subalgebra ofM(A).

Definition 9. Given two multipliers f₁, f₂ on A, we say thatf₁ extends f2 if dom(f2) ⊆ dom(f1) and f1(x) = f2(x), for all x ∈ dom(f2); we write f₂ ≤ f₁ if f₁ extends f₂. A multiplier f is called maximal if f can not be extended to a strictly larger domain.

Lemma15. (i) Iff1, f2∈M(A), f ∈Mr(A) and f ≤f1, f ≤f2,thenf1

and f₂ coincide on the dom(f₁)∩dom(f₂);

(ii)Every multiplierf ∈M_r(A) can be extended to a maximal multiplier.

More precisely, each principal multiplier fa witha∈Aand dom(fa)∈S(A)∩ R(A)can be uniquely extended to the total multiplierf_aand each non-principal multiplier can be extended to a maximal non-principal one.

Proof. (i) If there exists t∈dom(f₁)∩dom(f₂) such that f₁(t)6=f₂(t), since dom(f) ∈ R(A), then there exists t⁰ ∈ dom(f) such that t⁰ ∨f1(x) 6=

t⁰∨f₂(x)⇔f₁(t⁰∨t)6=f₂(t⁰∨t), which is contradictory sincet⁰∨t∈dom(f).

(ii) We first prove that fa cannot be extended to a non-principal multiplier. Let D= dom(f_a)∈S(A)∩R(A), f_a :D→A and suppose by contrary that there exists D⁰ ∈ S(A), D⊆D⁰ (hence D⁰ ∈R(A)) and a non-principal

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multiplier f ∈M(D⁰, A) which extendsfa.Since f is non-principal, there exists x₀ ∈D⁰,x₀ ∈/ D such that f(x₀) 6=a∨x₀. Since D∈R(A),there exists t ∈D such that t∨f(x0) 6=t∨(a∨x0) ⇔f(t∨x0)6=a∨(t∨x0),which is contradictory since fa≤f. Hencefa is uniquely extended byfa.

Now, let f ∈ M_r(A) be non-principal and M_f = {(D, g) : D ∈ S(A), g ∈ M(D, A), dom(f) ⊆ D and g|dom(f) = f} (clearly, if (D, g) ∈ Mf, then D∈S(A)∩R(A)).

The setM_f is ordered by (D₁, g₁)≤(D₂, g₂) iffD₁ ⊆D₂ andg_2|D₁ =g₁. Let {(D_k, g_k) : k ∈ K} be a chain in M_f. Then D⁰ = S

k∈K

D_k ∈ S(A) and dom(f)⊆D⁰.So,g⁰ :D⁰ →A defined byg⁰(x) =g_k(x) ifx∈D_k is correctly defined (since if x∈Dk∩Dtwith k, t∈K,then by (i),gk(x) =gt(x)).

Clearly,g⁰ ∈M(D⁰, A) and g⁰_|dom(f) =f (since if x∈dom(f)⊆D⁰, then x∈D⁰ and so there existsk∈Ksuch thatx∈Dk,henceg⁰(x) =gk(x) =f(x)).

So, (D⁰, g⁰) is an upper bound for the family{(D_k, g_k) :k∈K},hence by Zorn’s lemma, M_f contains at least one maximal multiplier h which extends f. Since f is non-principal and hextends f, h is also non-principal.

On the Hilbert algebra M_r(A) we consider the relation ρ_A defined by (f1, f2)∈ρA ifff1 and f2 coincide on the intersection of their domains.

Lemma 16. ρ_A is a congruence on M_r(A).

Proof. The reflexivity and symmetry ofρAare immediately; to prove the transitivity of ρ_A let (f₁, f₂),(f₂, f₃)∈ ρ_A. Therefore, f₁, f₂ and respectively f₂, f₃ coincide on the intersection of their domains. If by contrary, there exists x0 ∈ dom(f1)∩dom(f3) such that f1(x0) 6= f3(x0), since dom(f2) ∈ R(A), there exists t ∈ dom(f₂) such that t∨f₁(x₀) 6= t∨f₃(x₀) ⇔ f₁(t∨x₀) 6=

f3(t∨x0) which is contradictory, sincet∨x0 ∈dom(f1)∩dom(f2)∩dom(f3).

The compatibility ofρ_A with→ on M_r(A) is immediate.

For f ∈ Mr(A) we denote by [f] the congruence class of f modulo ρA

and A⁰⁰=Mr(A)/ρ_A .

Lemma17. The mapv_A:A→A⁰⁰defined byv_A(a) = [f_a]is an injective morphism of Hilbert algebras and vA(A)∈R(A⁰⁰).

Proof. The fact thatv_Ais a morphism of Hilbert algebras follows from Lemma 12. To prove the injectivity ofv_Aleta, b∈Asuch thatv_A(a) =v_A(b).

Then [fa] = [fb]⇔(fa, fb) ∈ρA⇔fa(x) =fb(x),for every x∈A⇔x∨a= x∨b,for everyx∈A⇔a=b.

To prove vA(A) ∈ R(A⁰⁰), if by contrary there exist f1, f2 ∈ Mr(A) such that [f1]6= [f₂] (that is, there exists x0 ∈dom(f1)∩dom(f2) such that f₁(x₀) 6=f₂(x₀)) and [f₁]∨[f_a] = [f₂]∨[f_a]⇔ [f₁∨f_a] = [f₂∨f_a] for every a ∈ A, then for a = x0 we obtain that for every x ∈ dom(f1)∩dom(f2),

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(f1 ∨fx0)(x) = (f2 ∨fx0)(x) ⇔ x0 ∨x∨f1(x) = x0 ∨x∨f2(x). For x = x₀ we obtain that f₁(x₀)∨x₀ = f₂(x₀) ∨x₀ ⇔ f₁(x₀) = f₂(x₀), which is contradictory.

Remark 3. Since for every a ∈ A, f_a is the unique maximal multiplier on [fa] (by Lemma 15) we can identify [fa] with fa. So, since vA is injective morphism of Hilbert algebras, the elements of A can be identified with the elements of the set {f_a:a∈A}.

Lemma 18. In view of the identifications made above, if [f]∈A⁰⁰ (with f ∈M_r(A) and D= dom(f)∈S(A)∩R(A)),then

D⊆ {a∈A:f_a∨[f]∈A}.

Proof. Let a∈D.If by contrary, fa∨[f]∈/ A (that is [fa∨f]∈/ vA(A)), then f_a∨f is a non-principal multiplier.

Then by Lemma 15 (ii),f_a∨f can be extended to a non-principal maximal multiplier f : D → A with D ∈ S(A). Thus, D ⊆ D and for every x ∈ D, f(x) = (fa∨f)(x) = a∨x∨f(x) = a∨f(x). Since a ∈ D, then f(x) = f(a∨x) = x∨f(a), that is, f_|D is principal which is contradictory with the assumption that f is non-principal.

5. MAXIMAL COMMUTATIVE HILBERT ALGEBRA OF QUOTIENTS

The goal of this section is to define (taking as a guide-line the case of distributive lattices) the notions of Hilbert algebra of fractions and maximal Hilbert algebra of quotients for a commutative (unbounded) Hilbert algebra (for the case of bounded Hilbert algebras see [4]).

For some informal explanations of notions of fraction see [12, p. 37] for the case of rings.

Definition 10. A commutative Hilbert algebraA⁰is calledHilbert algebra of fractions of Aif

(a13) A is a Hilbert subalgebra ofA⁰;

(a₁₄) for everya⁰, b⁰, c⁰∈A⁰,a⁰6=b⁰,there existsa∈Asuch thata∨a⁰6=a∨b⁰ and a∨c⁰∈A.

So, every commutative Hilbert algebra is a Hilbert algebra of fractions of itself.

As a notational convenience, we write A ≺ A⁰ to indicate that A⁰ is a Hilbert algebra of fractions ofA.

Definition 11. Q(A) is the maximal (commutative) Hilbert algebra of quotients of A if A ≺ Q(A) and for every A⁰ with A ≺ A⁰ there exists a monomorphism of Hilbert algebras i:A⁰ →Q(A).

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Lemma 19. Let A ≺ A⁰; then for every a⁰, b⁰ ∈ A⁰, a⁰ 6= b⁰, and any finite sequence c⁰₁, . . . , c⁰_n∈A⁰, there existsa∈A such that a∨a⁰ 6=a∨b⁰ and a∨c⁰_i ∈A for i= 1,2, . . . , n, n≥2.

Proof. Assume lemma holds true for n−1.So we may find b∈ A such that b∨a⁰ 6= b∨b⁰ and b∨c⁰_i ∈ A for i = 1,2, . . . , n−1. Since A ≺ A⁰, we find c ∈ A such that c∨(b∨a⁰) 6= c∨(b∨b⁰) and c∨c⁰_n ∈ A. The element a=b∨c∈A has the required properties.

Lemma 20. Let A≺A⁰ anda⁰ ∈A⁰.Then D_a⁰ ={a∈A:a∨a⁰ ∈A} ∈ S(A)∩R(A).

Proof. If a∈ A and x ∈ D_a⁰, then x∨a⁰ ∈ A and since (a∨x)∨a⁰ = a∨(x∨a⁰)∈A it followsa∨x∈D_a⁰,hence D_a⁰ ∈S(A).

If on the contrary, x 6= y, since A ≺ A⁰, there exists a₀ ∈ A such that a0∨a⁰ ∈A(that is a0 ∈Da⁰) anda0∨x6=a0∨y,which is contradictory.

Theorem 21. A⁰⁰ (defined in Section 4) is the maximal (commutative) Hilbert algebra of quotients Q(A) of A.

Proof. The fact that A is a Hilbert subalgebra of Q(A) follows from Lemma 17 and Remark 3. To prove A ≺ Q(A), let [f],[g],[h] ∈ Q(A) with f, g, h∈M_r(A) such that [g]6= [h] (that is, there existsx₀ ∈dom(g)∩dom(h) such that g(x0)6=h(x0)).

PutD= dom(f)∈S(A)∩R(A) and D_[f]={a∈A:fa∨[f]∈A}.Then by Lemma 18, D⊆D_[f].

If we suppose that for every a∈ D, fa∨[g] = fa∨[h], then [fa∨g] = [f_a∨h], hence for everyx∈dom(g)∩dom(h) we have (f_a∨g)(x) = (f_a∨h)(x)

⇔(analogously as in the proof of Lemma 17)⇔ a∨x∨g(x) =a∨x∨h(x)⇔ a∨g(x) =a∨h(x).

Since D ∈ R(A) we deduce that g(x) = h(x) for every x ∈ dom(g)∩ dom(h), so [g] = [h], which is contradictory. Hence, if [g] 6= [h], then there exists a ∈ D, such that fa ∨[g] 6= fa ∨[h]. But for this a ∈ D we have f_a∨[f]∈A (sinceD⊆D_[f_]), hence A≺Q(A).

To prove the maximality ofQ(A),letA⁰ be a commutative Hilbert algebra such thatA≺A⁰; ThenA⁰ is embedded inQ(A) byi:A⁰ →Q(A) defined by i(a⁰) = [f_a⁰], for every a⁰ ∈ A⁰, where dom(f_a⁰) = D_a⁰ (see Lemma 20).

Clearly,f_a⁰ ∈Mr(A) andiis a morphism of Hilbert algebras (see Lemma 12).

To prove the injectivity of i, let a⁰, b⁰ ∈ A⁰ such that i(a⁰) = i(b⁰) ⇔ [f_a⁰] = [f_b⁰]⇔ f_a⁰(x) =f_b⁰(x) for everyx∈D_a⁰∩D_b⁰.If a⁰ 6=b⁰,by Lemma 19 (since A ≺ A⁰), there exists a ∈ A such that a∨a⁰, a∨b⁰ ∈ A and a∨a⁰ 6= a∨b⁰ which is contradictory (since a∨a⁰, a∨b⁰ ∈A impliesa∈Da⁰ ∩Db⁰).

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Remark 4. Following Remark 1, ifAis bounded, thenAis a Boole algebra. So, in this caseQ(A) is just the classical Dedekind-Mac Neille completion of A (see [13], p. 239).

6. LOCALIZATION OF COMMUTATIVE HILBERT ALGEBRAS

The concept of multiplier for distributive lattices was defined by Cor- nish [7]. Schmid [16], used the multipliers in order to give a non–standard construction of the maximal lattice of quotients for a distributive lattice. A direct treatment of the lattices of quotients can be found in [17].

G. Georgescu [10] exhibited the localization latticeLF of a distributive lattice Lwith respect to a topology F on Lin a similar way as for rings (see [14]) or monoids (see [18]).

The concept of Hilbert algebra of fractions relative to a∨-closed system, maximal Hilbert algebra of quotients and Hilbert algebra of localization was studied by Busneag [2]–[6] forbounded case (taking as a guide-line the case of distributive lattices).

The aim of this section is to define the notion of localization Hilbert algebra of a commutative (unbounded) Hilbert algebra. In the last part of this paper is proved that the maximal (commutative) Hilbert algebra of fractions (defined in Section 4) and the (commutative) Hilbert algebra of fractions relative to a ∨-closed system (defined in Section 3) are Hilbert algebras of localization.

In this section by A we denote a commutative (unbounded) Hilbert algebra and by S(A) the set {D ⊆A:a∈ A, x∈D ⇒a∨x ∈D}. We have, Ds(A) ⊆ S(A) and every D ∈ S(A) is a ∨-closed system of A. Clearly, if D∈S(A),then 1∈Dand if x≤y and x∈D,theny=x∨y∈D.

Definition 12. A non-empty family F of elements onS(A) will be called a topology on A when

(a₁₅) if D₁∈ F,D₂∈S(A) and D₁ ⊆D_2, thenD₂∈ F (hence A∈ F);

(a16) if D1, D2 ∈ F then D1∩D2 ∈ F.

Example 3. If D∈S(A) then the set F(D) ={D⁰ ∈S(A) :D⊆D⁰} is a topology on A.

Example 4. We recall that by R(A) we denote the set of all regular subsets of A (see Definition 8). Then F =S(A)∩R(A) is a topology on A, see Lemma 13.

Example 5. Let S ⊆ A a ∨-closed subset of A (see Definition 5). If we denote by F_S={D∈S(A) :D∩S6=},then F_S is a topology onA.

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In the following by A we denote a commutative (unbounded) Hilbert algebra and by F a topology on A. Let us consider the relation θF of A defined by

(x, y)∈θF ⇔there existsD∈ F such thatt∨x=t∨y for anyt∈D.

As in the case ofθ_S(see Proposition 5), we deduce thatθFis a congruence on A.We shall denote byx/θF the congruence class of an elementx∈A and by pF :A→A/θF the canonical morphism of Hilbert algebras.

Definition 13. A F-multiplier on A is a mappingf :D →A/θF where D∈ F such that

(a17) f(a∨x) =a/θF ∨f(x)

for every a∈A and x∈D. Clearly, x/θF ≤f(x) for everyx∈D.

IfF ={A},thenF-multiplier is a total multiplier in sense of Definition 7.

The maps0,1:A→A/θF defined by0(x) =x/θF and1(x) = 1/θF for every x∈D areF-multipliers.

Also, for a ∈ A, fa : D → A/θF defined by fa(x) = a/θF ∨x/θF for every x∈D,is aF-multiplier.

We shall denote by M(D, A/θF) the set of all the F-multipliers having the domain D ∈ F. If D1, D2 ∈ F, D1 ⊆ D2, we have a canonical mapping ϕ_D₁_,D₂ :M(D₂, A/θF) → M(D₁, A/θF) defined by ϕ_D₁_,D₂(f) =f_|D₁ forf ∈ M(D₂, A/θF).Let us consider the directed system of setsh{M(D, A/θF)}D∈F, {ϕ_D₁_,D₂}_D₁_,D₂_∈F,D₁_⊆D₂iand denote byAF the inductive limit (in the category of sets): AF = lim−→

D∈F

M(D, A/θF).

For anyF-multiplierf :D→A/θF we shall denote by(D, f\) the equivalence class of f inAF.

Remark 5. We recall that if fi : Di → A/θF, i= 1,2, are multipliers, then (D\₁, f₁) = (D\₂, f₂) (in AF) iff there exists D ∈ F, D ⊆D₁∩D₂, such that f1|D =f2|D.

Let f_i : D_i → A/θF, (with D_i ∈ F, i = 1,2), F-multipliers. Let us consider the mapping f1 →f2 :D1∩D2 → A/θF defined by (f1 → f2)(x) = f1(x)→f2(x),for any x∈D1∩D2,and let

(D\1, f1)7−→(D\2, f2) =(D1∩D\2, f1→f2).

This definition is correct. Indeed, let f_i⁰ : D_i⁰ → A/θF, with D_i⁰ ∈ F, i = 1,2, such that(D\_i, f_i) =(D\⁰_i, f_i⁰), i= 1,2. Then there exist D⁰⁰₁, D₂⁰⁰ ∈ F such thatD⁰⁰₁ ⊆D₁∩D⁰₁ and D₂⁰⁰⊆D₂∩D⁰₂ and f_1|D⁰⁰

1 =f_1|D⁰ 00 1, f_2|D⁰⁰

2 =f_2|D⁰ 00 2. If we set D⁰⁰ = D⁰⁰₁ ∩D⁰⁰₂ ⊆ D₁ ∩D₂ ∩D⁰₁ ∩D⁰₂, then D⁰⁰ ∈ F and clearly (f₁→f₂)_|D⁰⁰ = (f₁⁰ →f₂⁰)_|D⁰⁰,hence(D₁∩D\₂, f₁→f₂) =(D₁⁰ ∩D\₂⁰, f₁⁰ →f₂⁰).

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Lemma 22. f1→f2 ∈M(D1∩D2, A/θF).

Proof. If x ∈D₁∩D₂ and a∈A, then (f₁ → f₂)(a∨x) = f₁(a∨x) → f2(a∨x) = (a/θF ∨f1(x))→(a/θF ∨f2(x))^c=¹⁴a/θF∨(f1 →f2)(x).

Corollary 23. (AF,7→,0,1) is a bounded commutative Hilbert algebra (where 0=(A,[0) and 1=(A,[1)) (see Corollary 11).

Definition 14. AF will be called thelocalization Hilbert algebra of Awith respect to the topology F .

Lemma 24. The mapping vF :A → AF defined by vF(a) = (A, f\_a) for every a∈ A is a morphism of Hilbert algebras and vF(A) is a regular subset of AF.

Proof. If a, b∈Athen

vF(a)7→vF(b) =(A, f\a)7→(A, f\b) =(A, f\a→fb) =(A, f\a→b) =vF(a→b).

To prove thatvF(A) is a regular subset ofAF,let(D\_i, f_i)∈AF, D_i ∈ F, i= 1,2,such that(A, f\_a)∨(D\₁, f₁) =(A, f\_a)∨(D\₂, f₂) for everya∈A.

Then (D₁\, f_a∨f₁) = (D₂\, f_a∨f₂) ⇔ hence there exists D ⊆ D₁∩D₂, D∈ F such that (fa∨f₁)|D = (fa∨f₂)|D ⇔(a∨x)/θF∨f₁(x) = (a∨x)/θF∨f₂, for every x∈D anda∈A.

If in this last equivalence we choose a = x ∈ D, then we obtain that x/θF ∨f₁(x) = x/θF ∨f₂(x) ⇔ f₁(x) = f₂(x) ⇔ (D\₁, f₁) = (D\₂, f₂), hence vF(A) is a regular subset ofAF.

7. APPLICATIONS

In the following we describe the localization Hilbert algebraAF in some special instances.

1. If D ∈ S(A) and F is the topology F(D) = {D⁰ ∈ S(A) : D ⊆ D⁰} (see Section 4), then AF ⊆ M(D, A/θF) and vF : A → AF is defined by vF(a) =(D, f\_a|D) for any a∈A.

For x, y ∈ A we have (x, y) ∈ θF ⇔ for every t ∈ D, t∨x = t∨y ⇔ f_x|D = f_y|D ⇔ vF(x) = vF(y) then there exists an injective morphism of Hilbert algebras ϕ:A/θF →AF such that the diagram

A −→^v^F AF

&

pF

%

ϕ

A/θF

is commutative.

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2. If F = R(A) ∩S(A) then θF is the identity congruence of A and AF = lim−→

D∈F

M(D, A).

In this situation it is easy to see thatvF is injective, so we have

Proposition25. In the caseF =I(A)∩R(A), AF is exactly the maximal(commutative)Hilbert algebra of quotientsQ(A)ofA(see Section5, Theo- rem 21).

3. Let S be a ∨-closed system of A. We recall (see Proposition 5) that on A we have the congruenceθS defined by: (x, y)∈θS ⇔ there existss∈S such that s∨x =s∨y and A[S] = A/θ_S is called the (commutative) Hilbert algebra of fractions of A relative to the ∨-closed system S (see Section 3).

In this case we have the topology F_S associated with S,F_S = {D ∈ S(A) :D∩S 6=}.

Lemma 26. θFS =θS.

Proof. For x, y∈ A,if (x, y)∈ θF_S then there exists D ∈ F_S such that s∨x = s∨y for every s ∈ S. Since D∩S 6= there existss₀ ∈ D∩S; in particular, we obtain s0∨x=s0∨y,hence (x, y)∈θS,that is, θFS ⊆θS.

If (x, y)∈θ_Sthens₀∨x=s₀∨y,for somes₀ ∈S.If considerD= [s₀) (the principal deductive system generated bys0),thenD∈ F_S (sinces0∈D∩S).

If s∈Dthen s₀≤s⇒s=s∨s₀ hence s∨x= (s∨s₀)∨x=s∨(s₀∨x) = s∨(s₀∨y) = (s∨s₀)∨y=s∨y⇒(x, y)∈θF_S ⇒θ_S⊆θF_S ⇒θF_S =θ_S. Proposition 27. If F_S is the topology onA associated with a ∨-closed subset S of A, then the commutative Hilbert algebras A[S] and AF_S are iso- morphic.

Proof. By Lemma 26, θFS = θS, therefore a F_S-multiplier can be considered in this case as a mappingf :D→A[S] (D∈ F_S) having the property f(a∨x) =a/θ_S∨f(x),for everyx∈D and a∈A.

If (D\₁, f₁),(D\₂, f₂) ∈ AF_S = lim−→

D∈FS

M(D, A[S]), and (D\₁, f₁) = (D\₂, f₂) then there exists D ∈ F_S such that D ⊆ D1 ∩D2 and f1|D = f2|D. Since D, D₁, D₂ ∈ F_S,thenD∩S, D₁∩S, D₂∩S are nonempty, hence there exists∈ D∩S,s1 ∈D1∩Sands2 ∈D2∩S.We shall prove thatf1(s1) =f2(s2).Indeed, if consider t=s∨s1∨s2 ∈D∩S,thenf1(t) =s/θ_S∨s2/θ_S∨f1(s1) =f1(s1) (since s/θ_S =s₂/θ_S =0) and analogously f₂(t) =f₂(s₂) ⇒ f₁(s₁) =f₂(s₂).

In a similar way we can show that f1(s1) =f2(s2) for any s1, s2∈D∩S.

In accordance with these considerations we can define the mappingα : AF_S = lim−→

D∈F_S

M(D, A[S])→A[S] by puttingα((D, f\)) =f(s), wheres∈D∩S.

It is easy to prove that α is a morphism of Hilbert algebras.

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We shall prove thatαis injective and surjective. To prove the injectivity of αlet(D\1, f1),(D\2, f2)∈AF_S such thatα((D\1, f1)) =α((D\2, f2)).Then for anys1 ∈D1∩S, s₂ ∈D2∩S we havef1(s1) =f2(s2).We consider the element s= s₁∨s₂ ∈ (D₁ ∩D₂)∩S. We have f₁(s) = s₂/θ_S∨f₁(s₁) = 0∨f₁(s₁) = f1(s1) and f2(s) = s1/θS ∨f2(s2) = 0∨f2(s2), hence f1(s) = f2(s). Now let D_s = {s⁰ ∈ D₁ ∩D₂ : s ≤ s⁰}. Since s ∈ D_s we deduce that D_s 6= ∅.

If a ∈ A and s⁰ ∈ Ds then s ≤ s⁰ ≤ a∨s⁰ ⇒ a∨s⁰ ∈ Ds ⇒ Ds ∈ S(A).

Since s ∈ D_s∩D ⇒ D_s ∈ F_S. If s⁰ ∈ D_s, then s∨s⁰ = s⁰ ⇒ f₁(s⁰) = f1(s∨s⁰) =s⁰/θS∨f1(s) =0∨f1(s) =f1(s) and analogously,f2(s⁰) =f2(s)⇒ f_1|D_s =f_2|D_s ⇒ (D\₁, f₁) =(D\₂, f₂),that is, α is injective.

To prove the surjectivity of α, let a/S ∈ A[S] with a ∈ A. For s ∈ S we consider D= [s) (the principal deductive system generated by s).Clearly, D ∈ F_S. We define f_a :D → A[S] by putting f_a(x) = (a∨x)/θ_S, for every x ∈ D; fa is an F_S-multiplier. From (a∨s) ∨s = a∨s ⇒ (a∨s)/θS = a/θ_S ⇒ f_a(s)/θ_S =a/θ_S ⇒ α((D, f\_a)) =a/θ_S, that is, α is surjective, hence bijective.

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Received 13 March 2009 University of Craiova

Faculty of Mathematics and Computer Science 13, Al.I. Cuza st.

200585, Craiova, Romania [email protected] [email protected]