BRICS Basic Research in Computer Science

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B R ICS R S -02-23 Nola & L eu s¸tean : Comp act R ep re sen tation s of B L -Algeb ras

BRICS

Basic Research in Computer Science

Compact Representations of BL-Algebras

Antonio Di Nola Laurent¸iu Leus¸tean

BRICS Report Series RS-02-23

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Copyright c 2002, Antonio Di Nola & Laurent¸iu Leus¸tean.

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Compact representations of BL-algebras

Antonio Di Nola

^∗

and Laurent¸iu Leu¸stean

^∗∗

∗Universit`a di Salerno, Facolt`a di Scienze, Dipartimento di Matematica e Informatica,

Via S. Allende, 84081 Baronissi, Salerno, Italy E-mail: dinola@ds.unina.it

∗∗ National Institute for Research and Developement in Informatics,

8-10 Averescu Avenue, 71316, Bucharest, 1, Romania, E-mail: leo@u3.ici.ro

Abstract

In this paper we define sheaf spaces of BL-algebras (or BL-sheaf spaces), we study completely regular and compact BL-sheaf spaces and compact representations of BL-algebras and, finally, we prove that the category of non-trivial BL-algebras is equivalent with the category of compact local BL-sheaf spaces.

Introduction

BL-algebras are the algebraic structures for H´ajek’s Basic Logic [11]. The main example of a BL-algebra is the interval [0,1] endowed with the structure induced by at-norm.

In this paper we study compact representations of BL-algebras, following tech- niques used for ringed spaces by Mulvey [14, 13, 15]. In [13], Mulvey extended the concepts of complete regularity and compactness from topological spaces to ringed spaces and proved a compactness theorem for completely regular ringed spaces generalizing the Gelfand-Kolmogoroff criterion concerning maximal ide- als in the ringR(X) of continuous real functions on a completely regular space X[8]. In [14], Mulvey introduced compact representations of rings, showing that they are exactly those representations of rings that establish an equivalence of categories of modules. Using compact representations, Mulvey extended the Gelfand duality between the categories of compact spaces and commutativeC^∗- algebras to Gelfand rings [15].

Gelfand rings are characterized by a property that can be formulated in terms of universal algebra, namely that each prime ideal is contained in a unique maximal ideal. Universal algebras with this property and their Gelfand representations

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were studied by Georgescu and Voiculescu [10] and, in a lattice-theoretical set- ting, by Simmons [18].

MV-algebras [1] and BL-algebras are classes of algebras that also satisfy this property. Hence, the problem of obtaining similar results for these structures is natural. Filipoiu and Georgescu [7] proved that the category of MV-algebras is equivalent with the category of compact sheaf spaces of MV-algebras with local stalks.

In the present paper, we give an answer for this problem in the case of BL- algebras. In different classes of problems, sheaf representations of universal algebras are very useful since they reduce the study of algebras to the study of the stalks, which usually have a better known structure. In the case of our compact representations, the stalks are local BL-algebras, introduced and studied by Turunen and Sessa [22].

In the first section of the paper we recall some facts about BL-algebras and we study some special filters of BL-algebras, used in the sequel. In Section 2, we define sheaf spaces of BL-algebras (or BL-sheaf spaces), BL-algebras of global sections, morphisms of BL-sheaf spaces and other notions related with sheaf theory.

In the next section we define and study completely regular and compact BL- sheaf spaces and we prove the compactness theorem.

In the following section we remind some general results concerning sheaf representations of BL-algebras and we study a special kind of representations, namely compact representations. We prove that any compact representation of a BL- algebra arises canonically from a family of filters of the BL-algebra satisfying certain conditions.

Finally, in the last section of the paper we prove that the functor from the category of compact local BL-sheaf spaces to the category of non-trivial BL- algebras, obtained by assigning to each BL-sheaf space the BL-algebra of global sections determines an equivalence between these categories.

1 BL-algebras

ABL-algebra [11] is an algebra (A,∧,∨,,→,0,1) with four binary operations

∧,∨,,→and two constants 0,1 such that:

(i) (A,∧,∨,0,1) is a bounded lattice;

(ii) (A,,1) is a commutative monoid;

(iii)and→form an adjoint pair, i.e.

c≤a→b iffac≤bfor alla, b, c∈A;

(iv)a∧b=a(a→b);

(v) (a→b)∨(b→a) = 1.

A BL-algebraAis nontrivial iff 06= 1.

For any BL-algebraA, the reductL(A) = (A,∧,∨,0,1) is a bounded distributive lattice.

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A BL-chainis a linear BL-algebra, i.e. a BL-algebra such that its lattice order is total.

For anya∈A, we definea⁻ =a→0. We denote the set of natural numbers byω. We definea⁰= 1 andaⁿ=aⁿ⁻¹aforn∈ω− {0}. Theorderofa∈A, in symbolsord(a),is the smallestn∈ω such thataⁿ = 0. If no suchnexists, thenord(a) =∞.

The following properties hold in any BL-algebraAand will be used in the sequel:

(1.1) ab≤a∧b≤a, b (1.2) a≤b impliesac≤bc (1.3) 0→a= 1 and 1→a=a (1.4) a→b= 1 iffa≤b (1.5) ab= 0 iffa≤b⁻ (1.6) aa⁻= 0

(1.7) a→(b→c) = (ab)→c

(1.8) (a∧b)⁻=a⁻∨b⁻ and (a∨b)⁻=a⁻∧b⁻ (1.9) 1⁻= 0 and 0⁻= 1

(1.10) a⁻= 1 iffa= 0

(1.11)a∨b= 1 impliesaⁿ∨bⁿ= 1 for anyn∈ω

LetAbe a BL-algebra. AfilterofAis a nonempty setF ⊆Asuch that for all a, b∈A,

(i)a, b∈F impliesab∈F; (ii)a∈F anda≤bimplyb∈F. A filterF ofAisproperiffF 6=A.

By (1.1) it is obvious that any filter of Ais also a filter of the latticeL(A). A proper filterP ofAis calledprimeprovided that it is prime as a filter ofL(A):

a∨b∈P impliesa∈P orb∈P.

A proper filterM ofAis calledmaximal(orultrafilter) if it is not contained in any other proper filter.

We shall denote by Spec(A) the set of prime filters of A and by M ax(A) the set of maximal filters ofA. Let us remind some properties of filters that will be used in the sequel.

Proposition 1.1 ([21, Proposition 8]

IfA is a nontrivial BL-algebra, then any proper filter ofAcan be extended to a maximal filter.

Proposition 1.2 [21, Proposition 6]

LetP be a prime filter of a nontrivial BL-algebraA. Then the set F={F |P ⊆F andF is a proper filter ofA}

is linearly ordered with respect to set-theoretical inclusion.

Proposition 1.3 [6, Proposition 1.6]

If A is a nontrivial BL-algebra, then any prime filter of A is contained in a unique maximal filter.

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Proposition 1.4 [21, Proposition 7]

Any maximal filter ofAis a prime filter ofA.

LetX ⊆A. The filter ofAgenerated byX will be denoted by< X >. We have that<∅>={1}and< X >={a∈A|x₁ · · · xn≤afor somen∈ω− {0}

and some x₁,· · ·, xn ∈X} if∅ 6=X ⊆A. For anya ∈A, < a > denotes the principal filter ofAgenerated by{a}. Then, < a >={b∈A|aⁿ ≤bfor some n∈ω− {0}}.

Lemma 1.5 LetF, G be filters ofA. Then

< F∪G >={a∈A|bc≤afor some b∈F,c∈G}

Proposition 1.6 Let F(A) be the set of filters of A. Then (F(A),⊆) is a complete lattice. For every family{Fi}i∈I of filters ofA, we have that

∧i∈IFi=∩i∈IFi and∨i∈IFi =<∪i∈IFi> .

With any filter F of A we can associate a congruence relation ∼F on A by defining

a∼F b iffa→b∈F andb→a∈F iff (a→b)(b→a)∈F.

For anya∈A, leta/F be the equivalence classa/∼F. If we denote byA/F the quotient setA/_∼_F, thenA/F becomes a BL-algebra with the natural operations induced from those ofA.

Proposition 1.7 [11]

LetF be a filter ofAanda, b∈A.

(i)a/F = 1/F iffa∈F; (ii)a/F = 0/F iffa⁻ ∈F; (iii) for alla, b∈A,

a/F ≤b/F iffa→b∈F;

(iv)A/F is a BL-chain iffF is prime.

Ifh:A→B is a BL-morphism, then thekernelof his the setKer(h) ={a∈ A|h(a) = 1}. It is easy to see that

Proposition 1.8 Leth:A→B be a BL-morphism. IfGis a (proper, prime) filter of B, then h⁻¹(G) is a (proper, prime) filter of A. Thus, in particular, Ker(h) is a proper filter ofA.

Lemma 1.9 [5, Proposition 1.13]

LetAbe a nontrivial BL-algebra andM a proper filter ofA. The following are equivalent:

(i)M is maximal;

(ii) for anyx∈A,

x∈/M implies (xⁿ)⁻∈M for somen∈ω.

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Proposition 1.10 Leth:A→B be a BL-morphism. IfN is a maximal filter ofB, then h⁻¹(N) is a maximal filter ofA.

Proof: By Proposition 1.8, we have thath⁻¹(N) is a proper filter ofA. In order to get that it is maximal, we shall apply Lemma 1.9. Letx∈Asuch that x∈/h⁻¹(N), hence h(x)∈/N. Since N is a maximal filter of B, there isn ∈ω such that (h(x)ⁿ)⁻∈N, that is h((xⁿ)⁻)∈N, sincehis a homomorphism of BL-algebras. We have got that (xⁿ)⁻∈h⁻¹(N). 2

For any filterF ofA, let us denote by []F the natural homomorphism from A ontoA/F, defined by []F(a) =a/F for anya∈A. ThenF =Ker([]F).

LetAbe a BL-algebra andF a filter ofA.

(i) the map G 7→^α []F(G) is an inclusion-preserving bijective correspondence between the filters ofA containingF and the filters ofA/F. The inverse map is also inclusion-preserving;

(ii) Gis a proper filter of A containing F iff []F(G) is a proper filter of A/F. Hence, there is a bijection between the proper filters ofAcontainingF and the proper filters ofA/F;

(iii) there is a bijection between the maximal filters ofAcontainingF and the maximal filters ofA/F.

Following [22], a BL-algebraA islocal if it has a unique maximal filter.

Proposition 1.12 [22] LetAbe a local BL-algebra. Then its unique maximal filter is

{a∈A|ord(a) =∞}.

Proposition 1.13 LetP be a proper filter ofA. The following are equivalent:

(i)A/P is a local BL-algebra;

(ii)P is contained in a unique maximal filter ofA.

Proof: Apply [9, Proposition 2.6 ], and [9, Proposition 2.8 ]. 2

LetAbe a nontrivial BL-algebra. The prime spectrumofA is the setSpec(A) of prime filters ofA, endowed with the Zariski topoloy, of which the subsets of the form

D(a) ={P ∈Spec(A)|a∈/P} fora∈A form a basis of open sets.

Themaximal spectrum ofA is the subspaceM ax(A) ofSpec(A) consisting of the maximal filters ofAwith the induced topology. The subsets

d(a) =D(A)∩M ax(A) ={M ∈M ax(A)|a∈/M},a∈A

form a basis for the topology of the maximal spectrum. Then Spec(A) is a compact topological space andM ax(A) is compact and Hausdorff [12].

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In the sequel, we shall remind some facts concerning the reticulation of a BL- algebraA. For details see [12].

Let us define a binary relation≡onAby a≡b iffD(a) =D(b).

Then ≡ is an equivalence relation on A compatible with the operations,∧ and∨. Fora∈Alet us denote by [a] the class ofa∈Awith respect to≡. The bounded distributive latticeβ(A) = (A/≡,∨,∧,[0],[1]) is called thereticulation of the BL-algebraA.

If h : A →B is a homomorphism of BL-algebras, then β(h) : β(A) → β(B), defined byβ(h)(a) = [h(a)], is a homomorphism of bounded distributive lattices.

It follows that we can define a functor β from the category of nontrivial BL- algebras to the category of bounded distributive lattices. The functorβis called thereticulation functor.

IfF is a filter ofA, thenβ(F) ={[a]|a∈A} is a filter of the latticeβ(A) and the mapping F 7→β(F) is an isomorphism between the lattice F(A) of filters ofAand the latticeF(β(A)) of filters ofβ(A). IfP∈Spec(A), then β(P) is a prime filter ofβ(A) and the mappingP 7→β(P) is a homeomorphism between Spec(A) andSpec(β(A)). Similarly,M ax(A) is homeomorphic toM ax(β(A)).

Let us remind that a bounded distributive lattice Lis callednormal([23], [2]) if each prime ideal ofLcontains a unique minimal prime ideal.

For any nontrivial BL-algebraA, β(A) is a normal lattice.

To any prime filter P of a bounded distributive lattice or a BL-algebraA we associate the set

O(P) ={a∈A|a∨b= 1 for someb∈/P}.

Then it is easy to see thatO(P) is a proper filter ofAsuch thatO(P)⊆P. We have the following characterization of normal lattices

Proposition 1.15 [16, Theorem 3]

LetLbe a bounded distributive lattice. The following are equivalent:

(i)Lis normal;

(ii) for any maximal filterM ofL,M is the unique maximal filter that contains O(M).

Lemma 1.16 For any maximal filterM ofA, β(O(M)) =O(β(M)).

Proof: In the proof, we use that for alla∈A, [a] = [1] iffa= 1 and for each maximal filterM ofA,a∈M iff [a]∈β(M) [12].

“⊆” Let [a] ∈ β(O(M)), so there is b ∈ O(M) such that [a] = [b]. Since b∈O(M), there is c∈/Msuch thatb∨c= 1. It follows that [a]∨[c] = [b]∨[c] = [1]

and [c]∈/β(M). Hence, [a]∈O(β(M).

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“⊇” If [a]∈O(β(M)), then there is [b]∈/β(M) such that [a]∨[b] = [1]. Hence, there isb∈/M such thata∨b= 1, that isa∈O(M), so [a]∈β(O(M)). 2.

Proposition 1.17 LetAbe a nontrivial BL-algebra. Then

(i) for any maximal filterM ofA,M is the unique maximal filter that contains O(M);

(ii) for any distinct maximal filtersM, N ofA,O(M)∨O(N) =A;

(iii)A/O(M) is local for any M ∈M ax(A).

Proof: (i) Apply Proposition 1.15, Lemma 1.16 and the properties of the reticulation ofA.

(ii) Suppose thatO(M)∨O(N) is a proper filter ofA. Then, using Proposition 1.1, we get a contradiction to (i).

(iii) Apply (i) and Proposition 1.13. 2 Proposition 1.18 [4, Proposition 4.36]

LetTAbe a nontrivial BL-algebra. Then

M∈Max(A)O(M) ={1}.

2 BL-sheaf spaces. Definitions and first proper- ties

Asheaf space of BL-algebras(or aBL-sheaf space) is a triple (F, p, X) such that the following properties are satisfied:

(i)F andX are topological spaces;

(ii)p:F →X is a local homeomorphism from F onto X;

(iii) for each x∈X,p⁻¹({x}) =Fx is a nontrivial BL-algebra with operations denoted by∨x,∧x,x,→x,0x,1x;

(iv) the functions (a, b)7→a∨xb,(a, b)7→a∧xb,(a, b)7→axb,(a, b)7→a→xb from the set {(a, b) ∈ F ×F | p(a) = p(b)} into F are continuous, where x=p(a) =p(b);

(v) the functions 0,1 :X→F, which assign to eachxinX the zero 0xand the unit 1x ofFx respectively, are continuous.

X is known as thebase space,F as thetotal space andFx is called thestalk of F atx∈X.

If Y ⊆X, then asection σ overY is a continuous mapσ :Y → F satisfying (p◦σ)(y) =yfor ally∈Y. The set of all sections overY form a nontrivial BL- algebra with the operations defined pointwise, that will be denoted by Γ(Y, F).

The elements of Γ(X, F) are calledglobal sections.

For everyσ, τ ∈Γ(Y, F), we shall use the following notation:

[σ=τ] ={y∈Y |σ(y) =τ(y)}.

A BL-sheaf space (F, p, X) is calledlocalif for eachx∈X the stalkFxis a local BL-algebra.

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We shall use the expressiona BL-algebra of global sectionsto refer to any BL- subalgebra of Γ(X, F). If A is a BL-algebra of global sections, then for each x∈X, we definepÂ_x :A→Fx bypÂ_x(σ) =σ(x) for allσ∈A. If A= Γ(X, F), then we shall denotepÂ_x bypx.

The following properties are well-known and will be used in the sequel. For details see [20, 3, 19].

Proposition 2.1 Let (F, p, X) be a BL-sheaf space.

(i) for anyY ⊆X andσ, τ ∈Γ(Y, F), the subset [σ=τ] is open inY;

(ii) for eacha∈F there are an open subsetU ofX and a sectionσ∈Γ(U, F) such thatp(a)∈U andσ(p(a)) =a;

(iii) ifZ⊆Y ⊆X andσ∈Γ(Y, F), thenσ|Z ∈Γ(Z, F);

(iv) the family{σ(U)|U is open inX, σ∈Γ(U, F)}is a basis for the topology ofF;

(v) ifA is a BL-algebra of global sections, thenp^A_x is a BL-morphism for each x∈X;

(vi) if (F, p, X) and (G, q, X) are BL-sheaf spaces and f : F → G such that q◦f =p, then

f is continuous ifff is open ifff is a local homeomorphism.

If A is a BL-algebra of global sections, U is an open subset of X and σ is a section overU, we say thatσislocally in the BL-algebra of global sectionsAif (*) there are an open covering (Ui)i∈I ofU and a family (σi)i∈I of elements of Asuch thatσ|U_i=σi|U_i for alli∈I.

The following lemma follows immediately from Proposition 2.1(iv).

Lemma 2.2 Let (F, p, X) be a BL-sheaf space and A a BL-algebra of global sections such that every section over an open subset of X is locally in the BL- algebraA. Then the family{σ(U)|U is open in X, σ∈A} is a basis for the topology ofF.

Proposition 2.3 Let (F, p, X) be a BL-sheaf space and A a BL-algebra of global sections. The following are equivalent:

(i) every section over an open subset ofX is locally in the BL-algebraA;

(ii) for eachx∈X, the BL-morphismp^A_x is onto.

Proof: (i)⇒(ii) Let x ∈ X and a ∈ Fx, that is a ∈ F such that p(a) = x.

Applying Proposition 2.1(ii), there is an open neighbourhood U of x and a section σover U such that σ(x) =a. By (i), we get an open covering (Ui)i∈I

of U and a family (σi)i∈I of sections fromA such that σ|U_i =σ−i|U_i for all i ∈ I. Since x ∈ U, we have that x ∈ Uk for some k ∈ I. It follows that σk(x) =σ(x) = a. Hence, we have gotσk ∈Asuch thatp^A_x(σk) =a. That is, p^A_x is onto.

(ii)⇒(i) LetU be an open subset ofX andσa section overU. For eachx∈U,

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we have thatσ(x)∈Fx, hence, by (ii), there isτ^x∈A such thatτ^x(x) =σ(x).

Let (F, p, X) be a BL-sheaf space andσ∈Γ(Y, F) a section overY ⊆X. The cosupportofσ,cosupp(σ), is the closed hull in the subspaceY of the set of those pointsx∈Y for whichσ(x)6= 1x:

cosupp(σ) ={x∈Y |σ(x)6= 1x}.

It is easy to see that (cosupp(σ))^c= [σ= 1|Y].

LetX andY be topological spaces andf :Y →X a continuous function. Let (F, p, X) and (G, q, Y) be two BL-sheaf spaces. A morphism α: F →G over f is a family (αy : F_f₍_y₎ → Gy)y∈Y of BL-morphisms satisfying the following condition:

If U is open in X and σ ∈ Γ(U, F), define β : f⁻¹(U) → G by β(y) = αy(σ(f(y))). Thenβ is continuous, and thereforeβ ∈Γ(f⁻¹(U), G). We shall writeβ =α^U_#(σ).

It follows that a morphism α : F → G over f induces a BL-morphism α^U_# : Γ(U, F)→Γ(f⁻¹(U), G) for all openU inX. We shall denoteα^X_# byϕ_#. Since f⁻¹(X) =Y,α_#is a BL-morphism between the BL-algebras of global sections Γ(X, F) and Γ(Y, G).

An example of a morphism over f is given by the canonical mapping from a BL-sheaf space (F, p, X) to the BL-sheaf space (f⁻¹(F), q, Y),inducedbyf and (F, p, X), defined as follows.

Define f⁻¹(F) = {(y, a) ∈ Y ×F | f(y) = p(a)} = S

y∈Y{y} ×Ff(y) and q:f⁻¹(F)→Y byq(y, a) =y. Then for ally∈Y,f⁻¹(F)y={y} ×Ff(y). For eachy∈Y, defineiy:Ff(y)→f⁻¹(F)y byiy(a) = (y, a). We get easily thatiy

is a bijection. We makef⁻¹(F)ya BL-algebra by transporting the BL-structure ofFf(y)to f⁻¹(F)y by means ofiy.

Thus, we have got a BL-sheaf space (f⁻¹(F), q, Y) and a morphismi : F → f⁻¹(F) overf, where iis the family (iy)y∈Y.

A morphism of BL-sheaf spaces(f, α) : (F, p, X)→(G, q, Y) consists of a continuous function f:Y →X and a morphismα:F →Goverf.

Anisomorphism of BL-sheaf spacesis a morphism (f, α) such thatf is a homeomorphism andαy is an isomorphism of BL-algebras for ally∈Y.

If (f, α) : (F, p, X) →(G, q, Y) and (g, β) : (G, q, y)→ (H, r, Z) are two morphisms of Bl-sheaf spaces, then their composition is the morphism (f◦g, β◦α), where (β◦α)z=βz◦αg(z)for allz∈Z.

Let (F, p, X) and (G, q, X) be BL-sheaf spaces over the same topological space X. If (αx :Fx →Gx)x∈X is a family of functions, then we can define a function α : F → G by α(a) = αx(a), wherex ∈ X is unique such that a∈ Fx.

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Conversely, a functionα:F →Gcan be seen as a family (αx:Fx→Gx)x∈X, whereαx=α|Fx for allx∈X.

Proposition 2.4 (1X, α) : (F, p, X) → (G, q, X) is a morphism of BL-sheaf spaces iffα:F→Gis a continuous function such thatq◦α=p andαx:Fx→ Gxis a BL-morphism for allx∈X.

Let us denote byBLthe category of nontrivial BL-algebras and BL-morphisms and byBL−ShSpthe category of BL-sheaf spaces and morphisms of BL-sheaf spaces.

DefineS(F, p, X) = Γ(X, F) for any BL-sheaf space (F, p, X) andS(f, α) =α_# for every morphism (f, α) : (F, p, X)→(G, q, Y). Then

Proposition 2.5 S:BL−ShSp→BLis a functor, called thesectionfunctor.

3 Compact BL-sheaf spaces

Throughout, BL-algebras are nontrivial and X will be assumed to denote a Hausdorff topological space.

A BL-sheaf space (F, p, X) is calledcompletely regularif it satisfies the following:

(CR) for each x∈ X and closed set C ⊆ X not containing x, there is σ ∈ Γ(X, F) such thatσ(x) = 0xandσ|C= 1|C.

A completely regular BL-sheaf space (F, p, X) is calledcompactif the topological spaceX is compact.

The following lemma gives equivalent characterizations of completely regular BL-sheaf spaces.

Lemma 3.1 Let (F, p, X) be a BL-sheaf space. The following are equivalent:

(i) (F, p, X) is completely regular;

(ii) for eachx∈X and every open neighbourhoodU ofxthere is σ∈Γ(X, F) such thatσ(x) = 0xandσ(y) = 1y for ally∈/U;

(iii) for eachx∈X and every open neighbourhoodU ofxthere isσ∈Γ(X, F) such thatσ(x) = 0xandcosupp(σ)⊆U.

Proof: (i)⇒(ii) LetC=U^c. ThenC is a closed subset ofX such thatx∈/C, and applying (i) we get (ii).

(ii)⇒(i) TakeU =C^c and apply (ii).

(ii)⇔(iii) Apply the fact that (cosupp(σ))^c= [σ= 1]. 2

Proposition 3.2 Let (F, p, X) be a completely regular BL-sheaf space. Then (i)X is a regular topological space;

(ii) every section over an open subset ofX is locally in the BL-algebra Γ(X, F) of global sections of the BL-sheaf space;

(iii) the family [σ= 1]σ∈Γ(X,F) form a basis for the topology ofX; (iv)Fx∼=A/Ker(px) for allx∈X.

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Proof: (i) Letx∈X andU an open neighbourhood ofx. Applying Lemma 3.1(iii), there isσ∈Γ(X, F) such that σ(x) = 0x and cosupp(σ)⊆U. Hence, x∈[σ= 0] and, since Fy is nontrivial for ally ∈X, we have that 0y 6= 1y for ally∈X, sox∈[σ= 0]⊆cosupp(σ). Hence, there is a closed neighbourhood C =cosupp(σ) ofx such that C ⊆U. Thus, the closed neighbourhoods of x form a basis for neighbourhoods, soX is regular.

(ii) We shall prove that there is satisfied (ii) from Proposition 2.3 with A = Γ(X, F). Hence, we have to show that for each x∈X,pxis onto. Let a∈Fx, that is a ∈ F such that p(a) = x. Applying Proposition 2.1(ii), there is an open neighbourhood U of x and a section τ over U such that τ(x) = a. By Lemma 3.1(iii), there is θ∈ Γ(X, F) such thatθ(x) = 0x andcosupp(θ)⊆U. Let σ : X → F defined by σ(y) = θ(y)⁻ →y τ(y) for y ∈ U and σ(y) = 1y

for y ∈/U. It is obvious that p◦σ = 1X and that px(σ) = σ(x) =θ(x)⁻ →x

τ(x) = 0⁻_x →x a = 1x → a = a. It remains to prove that σ is continuous.

Since cosupp(θ) ⊆ U, we get that U ∪(cosupp(θ))^c = X. Let us prove that σ(y) = 1y for all y ∈(cosupp(θ))^c. If y ∈/U, then σ(y) = 1y by the definition of σ. If y ∈ U ∩(cosupp(θ))^c, then θ(y) = 1y and σ(y) = θ(y)⁻ →y τ(y) = 1⁻_y →yτ(y) = 0y →τ(y) = 1y. Hence, we have got thatσ|U, σ|₍cosupp(θ))^c are continuous andU,(cosupp(θ))^c form an open covering ofX. It follows thatσis continuous. Thus, we have obtainedσ∈Γ(X, F) such thatpx(σ) =a.

(iii) We have that [σ= 1] is open inX for allσ∈Γ(X, F). We shall prove that for any x∈ X and any open neigbourhood U of x there is σ∈ Γ(X, F) such that x∈[σ= 1]⊆U. From this we get immediately that [σ= 1]σ∈Γ(X,F)form a basis for the topology of X. Applying Lemma 3.1(iii), there is τ ∈ Γ(X, F) such thatτ(x) = 0xandcosupp(τ)⊆U. Letσ=τ⁻. Then,σ(x) = (τ(x))⁻ = 0⁻_x = 1x, hencex∈[σ= 1]. Ify∈[σ= 1], thenσ(y) = 1y, that is (τ(y))⁻= 1y. It follows thatτ(y)6= 1y, since 0y 6= 1y, soy∈cosupp(τ)⊆U. Hence, we have proved that [σ= 1] is an open neighbourhood ofxcontained in U.

(iv) We have proved at (ii) that the BL-morphism px :A→Fx,px(σ) =σ(x) is onto. Hence,Fx∼=A/Ker(px). 2

Let A be a BL-algebra of global sections of the BL-sheaf space (F, p, X). We say thatAiscompletely regular in the BL-sheaf space(F, p, X) if for eachx∈X and closed setC⊆X not containingx, there isσ∈Asuch thatσ(x) = 0x and σ|C = 1|C.

If Ais completely regular in (F, p, X) andX is compact, then A is said to be compact in the BL-sheaf space(F, p, X).

It is easy to see that, as in Lemma 3.1,Ais completely regular in the BL-sheaf space (F, p, X) iff for eachx∈X and every open neighbourhood U of xthere isσ∈Asuch that σ(x) = 0x andσ(y) = 1y for ally ∈/U. The following result extends Proposition 3.2(i) and (iii) and its proof is similar.

Lemma 3.3 LetAbe a BL-algebra of global sections that is completely regular in (F, p, X). Then

(i)X is regular;

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(ii) the family [σ= 1]σ∈Aform a basis for the topology ofX.

The following lemma collects some obvious facts that will be used in the sequel.

Lemma 3.4 Let (F, p, X) be a BL-sheaf space.

(i) (F, p, X) is completely regular (compact) iff the BL-algebra Γ(X, F) of global sections is completely regular (compact) in (F, p, X);

(ii) Suppose thatAandB are BL-algebras of global sections such thatA⊆B. IfA is completely regular (compact) in (F, p, X), thenB is completely regular (compact) in (F, p, X);

(iii) If there is a BL-algebra A of global sections that is completely regular (compact) in (F, p, X), then (F, p, X) is completely regular (compact).

Proposition 3.5 Let A be a BL-algebra of global sections that is compact in (F, p, X) and suppose that every global section is locally in A. ThenA is necessarily the BL-algebra Γ(X, F).

Proof: Let σ ∈ Γ(X, F). Since σ is locally in A, it follows that there are an open covering (Ui)i∈I ofX and a family (σi)i∈I of elements ofAsuch that σ|U_i =σi|U_i for alli∈I. For eachx∈X, there isix∈Isuch thatx∈Ui_x and applying the fact thatA is completely regular in (F, p, X), we getτi_x ∈Asuch that τi_x(x) = 0x andτi_x(y) = 1y for all y ∈/Ui_x. Let us denoteUi_x∩[τi_x = 0]

by Vx. Then, x ∈ Vx ⊆Ui_x for all x∈ X, so the family (Vx)x∈X is an open covering of X. Since X is compact, it follows that there arex₁,· · ·, xn ∈ X such that X =Vx₁∪ · · · ∪Vx_n. Let us denoteVx_k byVk, ix_k byik andτi_xk by τk for allk = 1, n. We shall prove thatσ=V

k=1,n(σi_k∨τk). Letx∈X and J ={k= 1, n|x∈Ui_k}. It is obvious thatJ is nonempty, sinceS

k=1,nUi_k = X. We have that σi_k(x) = σ(x) for all k ∈ J and x ∈/Ui_k for all k ∈/J, so τk(x) = 1xfor allk∈/J. It follows that [V

k∈J(σi_k(x)∨τk(x))]∧[V

k∈/J(σi_k(x)∨ τk(x))] = [V

k∈J(σ(x)∨τk(x))]∧[V

k∈/J(σi_k(x)∨1x] =σ(x)∨V

k∈Jτk(x). Since X =S

k=1,nVk, there isj= 1, nsuch thatx∈Vj, soτj(x) = 0xandj∈J, since Vj⊆Ui_j. It follows that (V

k∈Jτk)(x) = 0x, hence [V

k=1,n(σi_k∨τk)](x) =σ(x).

Thus,σ=V

k=1,n(σi_k∨τk), hence σ∈A.

2

3.1 The compactness theorem

In the sequel, A will be a BL-algebra of global sections of the BL-sheaf space (F, p, X).

For eachx∈X, let us denoteKx=Ker(p^A_x) ={σ∈A|σ(x) = 1x}. SinceA is nontrivial, it follows thatKxis a proper filter ofA.

A filterT ofAis calledfixedif there isx∈X such thatT∨Kxis a proper filter ofA. Otherwise,T is said to be afreefilter ofA.

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Lemma 3.6 LetA be a BL-algebra of global sections of (F, p, X),P a prime filter andM a maximal filter ofA. Then

(i)M is fixed iffM contains the filterKxfor some x∈X;

(ii) ifMP is the unique maximal filter that contains P, thenP is fixed iffMP

is fixed;

(iii) ifP contains the filterKx for somex∈X, thenP is fixed.

Proof: (i) Suppose that M is fixed, so there isx∈ X such that M ∨Kx is a proper filter of A. SinceM ⊆ M ∨Kx and M is maximal, it follows that M ∨Kx=M, henceKx⊆M. Conversely, ifKx⊆M for somex∈X, we get that M∨Kx=M, soM∨Kxis a proper filter ofA. That is, M is fixed.

(ii) If MP is fixed, then, by (i), there is x ∈ X such that Kx ⊆ MP. Since P ⊆MP, we have that P ∨Kx ⊆MP, hence P∨Kx is a proper filter ofA, i.e. P is fixed. Conversely, suppose that P is fixed, that isP ∨Kx is proper for some x∈X. We get thatMP andP∨Kx are proper filters containing the prime filterP, so applying Proposition 1.2 and the fact thatMP is maximal, it follows thatP∨Kx⊆MP. Hence, Kx⊆MP, so by (i), MP is fixed.

(iii) Since Kx⊆MP, we get that MP is fixed, by (i). Applying (ii), we obtain that P is also fixed. 2

Lemma 3.7 LetAbe a BL-algebra of global sections of (F, p, X). The following are equivalent

(i) every proper filter ofAis fixed;

(ii) every prime filter ofAis fixed;

(iii) every maximal filter ofA is fixed.

Proof: (i)⇒(ii) Obviously.

(ii)⇒(iii) Apply the fact thatM ax(A)⊆Spec(A), by Proposition 1.4.

(iii)⇒(i) LetF be a proper filter ofA. By Proposition 1.1, there is a maximal filterM such thatF ⊆M. SinceM is fixed, we getx∈X such thatKx⊆M. We have thatF, Kx⊆M, soF∨Kx⊆M. Hence,F∨Kxis a proper filter of A, that isF is fixed. 2

Lemma 3.8 LetAbe a BL-algebra of global sections of (F, p, X) and suppose that X is compact. Then

(i) for every prime filterP ofAthere is x∈X such thatKx⊆P; (ii) every proper filter ofAis fixed.

Proof: (i) LetPbe a prime filter ofAand suppose thatKx⊆/P for anyx∈X. That is for any x∈X there isσ^x∈Kx such thatσ^x ∈/P. Since σ^x ∈Kx, we get that σ^x(x) = 1x, that isx∈[σ^x= 1]. Thus,X =S

x∈X[σ^x= 1], hence the family [σ^x= 1]x∈Xis an open covering ofX. SinceXis compact, it follows that there arex₁,· · ·, xn ∈X such thatX =Sn

i=1[σi= 1], whereσi denotes σ^xⁱ for i= 1, n. It follows immediately thatσ₁∨ · · · ∨σn = 1∈P. Since P is prime, we obtain thatσi ∈P for somei= 1, n. Thus, we have got a contradiction.

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(ii) Applying (i) and Lemma 3.6(iii), we obtain that every prime filter of A is fixed. Now apply Lemma 3.7 to get that every proper filter ofAis fixed. 2 In the following, we shall denote bySpecX(A) the set of prime filters ofAthat are fixed and byM axX(A) the set of maximal filters ofAthat are fixed.

Lemma 3.9 Suppose that Ais completely regular in (F, p, X). Then

(i) for anyP ∈SpecX(A) there is a uniquex∈X such thatKx⊆MP, where MP is the unique maximal filter that containsP;

(ii) for anyM ∈M axX(A) there is a unique x∈X such thatKx⊆M. Proof: (i) The existence ofx∈X such thatKx⊆MP follows from Lemma 3.6. It remains to prove the unicity. Let us suppose that there isy6=xsuch that Ky ⊆MP. SinceX is Hausdorff, there is an open neighbourhoodU ofxsuch that y ∈/U. Applying now Lemma 3.1(ii), there is σ∈A such thatσ(x) = 0x

and σ(z) = 1z for all z ∈/U. It follows that σ(y) = 1y, so σ ∈Ky ⊆MP and σ⁻(x) = 1x, hence σ⁻ ∈ Kx ⊆ MP. We have got that σ, σ⁻ ∈ MP, hence σσ⁻ = 0∈ MP. Thus, we have obtained that MP is not proper, that is a contradiction.

(ii) By (i). 2

IfAis completely regular in (F, p, X), then, by the above lemma, we can define a function s:SpecX(A)→ X that assigns to eachP ∈ SpecX(A) the unique x∈X such thatKx⊆MP. We shall denote bymits restriction toM axX(A).

Then m assigns to every fixed maximal filter M of A the unique x∈ X such that Kx⊆M.

Corollary 3.10 LetAbe a BL-algebra of global sections of (F, p, X) and suppose thatX is compact. Then for every prime filter P of Athere is a unique x∈X such thatKx⊆P.

Proof: Apply Lemmas 3.8 and 3.9. 2

Lemma 3.11 Suppose thatAis completely regular in (F, p, X). Then for any M ∈M axX(A),K_m(M)⊆O(M)

Proof: Letx=m(M) andσ ∈Kx. We get thatσ(x) = 1x, sox∈[σ= 1].

Applying the fact that Ais completely regular in (F, p, X), we getτ ∈A such that τ(x) = 0x and τ(y) = 1y for ally ∈/[σ = 1]. It is clear that σ∨τ = 1.

¿From τ(x) = 0x, it follows that τ⁻(x) = 1x, soτ⁻ ∈ Kx ⊆ M. Since M is proper, we must haveτ∈/M. Hence, there isτ∈/M such thatσ∨τ= 1, that is σ∈O(M). 2

Lemma 3.12 Let (F, p, X) be a completely regular local BL-sheaf space and A= Γ(X, F). Then

(i) for anyx∈X there is a unique M ∈M ax(A) such thatKx⊆M; (ii)K_m(_M₎=O(M) for anyM ∈M axX(A).

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Proof: (i) By Proposition 3.2(v) and the fact thatKer(px) =Kx, it follows that Fx ∼=A/Kx for allx ∈X. Hence, A/Kx is local for any x∈ X. Apply now Proposition 1.13.

(ii) Applying Proposition 3.11, we have thatK_m(M)⊆O(M). Let us prove the converse inclusion. If we denote x =m(M), then Kx ⊆M. Letσ ∈ O(M), so there is τ ∈/M such that σ∨τ = 1. SinceFx is local, its unique maximal filter is Nx = {a ∈ Fx | ord(a) = ∞}. By Proposition 1.10, we have that p⁻¹_x (Nx) is a maximal filter ofAand it is easy to see thatKx⊆p⁻¹_x (Nx). Since Kx⊆p⁻¹_x (Nx),Kx⊆M andp⁻¹_x (Nx), M are maximal filters ofA, applying (i) it follows thatp⁻¹_x (Nx) =M. Now,τ∈/M impliesτ ∈/p⁻¹_x (Nx), soord(τ(x)))<

∞. Thus, there isn∈ω− {0}such that (τ(x))ⁿ = 0x. Sinceσ∨τ= 1, we get thatσ(x)∨xτ(x) = 1x, so (σ(x))ⁿ∨x(τ(x))ⁿ = 1x, that is (σ(x))ⁿ = 1x, hence σ(x) = 1x. Thus, we have got that σ∈Kx. 2

Proposition 3.13 Let A be completely regular in (F, p, X). Then s is onto andm is continuous and onto.

Proof: Letx∈ X. Then Kx is a proper filter ofA, so, by Proposition 1.1, there is a maximal filter M such that Kx ⊆ M. Applying Lemma 3.6(i), we get thatM is fixed. Hence,M ∈M axX(A) is such thatm(M) =x. Thus,m is onto and, obviously, sis also onto. Let us prove now thatm is continuous.

LetM ∈M axX(A),x=m(M) andU an open neighbourhood ofx. SinceAis completely regular in (F, p, X)), there isσ∈Asuch thatσ(x) = 0xandσ(y) = 1y for all y∈/U. LetV =d(σ)∩M axX(A) ={N ∈M axX(A)|σ∈/N}. Then V is an open subset of M axX(A). Since σ(x) = 0x, we get that σ⁻(x) = 1x, that isσ⁻ ∈Kx⊆M. It follows that σ∈/M, henceM ∈V. Let us prove that m(V)⊆U. LetN ∈V andy =m(N), soKy ⊆N. Ify ∈/U, thenσ(y) = 1y, so σ ∈Ky, henceσ∈ N. This contradicts the fact that N ∈d(σ). It follows thaty∈U. Thus, we have proved thatV is an open neighbourhood ofM such that m(V)⊆U. That is, mis continuous atM. 2

Suppose that A is compact in (F, p, X). Then, by Lemma 3.8, we have that SpecX(A) =Spec(A) and, by Corollary 3.10,s:Spec(A)→X assigns to every prime filterPofAthe uniquex∈X such thatKx⊆P. We obtain the following corollary.

Corollary 3.14 LetAbe compact in the BL-sheaf space (F, p, X). Thensand mare continuous, closed and onto.

Proof: We get that s is continuous in a similar manner with the proof of continuity ofmfrom Proposition 3.13 . To obtain that the functions are closed, apply [17, Theorem 7.2.2, p. 71], sinces,m are continuous and onto,M ax(A) andSpec(A) are compact andX is Hausdorff. 2

Theorem 3.15 (The compactness theorem)

Suppose that A is completely regular in the BL-sheaf space (F, p, X). The

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following are equivalent

(i) the topological spaceX is compact;

(ii) every proper filter ofAis fixed;

(iii) every maximal filter of Ais fixed;

(iv) every prime filter ofA is fixed;

(v) Ais compact in the BL-sheaf space (F, p, X).

Proof: (i)⇔(v) By definition.

(ii)⇔(iii)⇔(iv) By Lemma 3.7.

(i)⇒(ii) Apply Lemma 3.8.

(ii)⇒(i) We have that M axX(A) = M ax(A) and m : M ax(A) → X. Since m is continuous and onto andM ax(A) is compact, applying a known result of topology, it follows thatX is also compact (see, e.g., [17, Theorem 7.2.1, p.71]).

2

Proposition 3.16 If (F, p, X) is a compact BL-sheaf space andA= Γ(X, F), then

mis a homeomorphism iff (F, p, X) is a local BL-sheaf space.

Proof: Applying Propositions 1.13 and 3.2(iv), it follows that m is injective iff for anyx∈X there is a unique maximal filterM ofAsuch thatm(M) =x iff for anyx∈X there is a unique maximal filterM ofAsuch thatKx⊆M iff for allx∈X,A/Kxis local iff for allx∈X,Fx is a local BL-algebra. Hence, if mis a homeomorphism, then (F, p, X) is a local BL-sheaf space. Conversely, if (F, p, X) is local, thenm is injective. We have that mis bijective, continuous and closed, by Corollary 3.14. Hence,m is a homeomorphism. 2

Let (F, p, X) be a compact local BL-sheaf space and A = Γ(X, F). By the proof of the above proposition, we can define a function n : X → M ax(A), that associates with every x∈X the unique maximal filterM ofA such that Kx⊆M. It is easy to see that

Proposition 3.17 Let (F, p, X) be a compact local BL-sheaf space. Thennis the inverse ofm, hencen:X →M ax(A) is also a homeomorphism.

4 Compact representations of BL-algebras

In the sequel,Awill be a nontrivial BL-algebra andX will be assumed to denote a Hausdorff topological space.

By a sheaf representation (or simply representation) of the BL-algebra A will be meant a BL-morphism

ϕ:A→Γ(X, F)

fromA to the BL-algebra of global sections of a BL-sheaf space (F, p, X).

Hence,ϕ(A) is a BL-algebra of global sections of (F, p, X). In a representation

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ϕ, eacha∈Adetermines a global sectionϕ(a); in particular, for everyx∈X, ϕ(a)(x) is an element of the stalk Fx.

For eachx∈X, we define

ϕx:A→Fx, ϕx(a) =ϕ(a)(x) for alla∈A, Kx=Ker(ϕx) ={a∈A|ϕ(a)(x) = 1x}.

Sinceϕx=px◦ϕ, we have thatϕx is a BL-morphism, so Kx is a proper filter ofAfor everyx∈X.

It is easy to see that Ker(ϕ) = T

x∈XKx, hence ϕ is a monomorphism iff T

x∈XKx={1}.

For everya∈A, we shall use the following notation:

V(a) = [ϕ(a) = 1] ={x∈X |ϕ(a)(x) = 1x}={x∈X |a∈Kx}.

By Proposition 2.1(i),V(a) is open inX for alla∈A.

A filter space of a BL-algebra A is a family (Tx)x∈X of proper filters of A, indexed by a Hausdorff topological spaceX.

Letϕ :A →Γ(X, F) be a representation ofA. The filter space (Kx)x∈X will be called the representation spaceof the representation, and the filters indexed the representation filters. The topology generated by the family (V(a))a∈A of subsets of X is called the representation topology on the space X. Then, any topology onX contains the representation topology.

We say that a filter space (Tx)x∈X canonically determinesa representation ofA if there is a representationϕ:A→Γ(X, F) such thatTx=Kxfor allx∈X. In the sequel, we shall give an existence theorem for representations of BL- algebras.

Let A be a nontrivial BL-algebra and (Tx)x∈X a filter space of A such that the subset V(a) = {x ∈ X | a ∈ Tx} is open in X for all a ∈ A. Then a BL-sheaf space (FA, pA, X) and a representation ϕ : A → Γ(X, FA) can be constructed in the following way, given in [3] for universal algebra. Let FA

be the disjoint union of the sets (A/Tx)x∈X and pA : FA → X the canonical projection, so p⁻¹_A ({x}) = A/Tx for allx∈X. For allx∈ X, Tx is a proper filter of A, so A/Tx is a nontrivial BL-algebra. For each a ∈ A, define the map [a] :X →FA by [a](x) =a/Tx. EndowFA with the topology generated by the family {[a](U) | a ∈ A and U is open in X}. Applying [3, Corollary 2], we get that (FA, pA, X) is a sheaf space of BL-algebras and the function ϕ:A→Γ(X, FA), defined byϕ(a) = [a] for alla∈A, is a representation ofA.

It is easy to see thatKx=Txfor allx∈X. Hence, we get the following theorem:

Theorem 4.1 LetAbe a nontrivial BL-algebra and (Tx)x∈X a filter space of A such that the subset V(a) ={x∈X | a∈Tx} is open in X for all a∈A.

Then (Tx)x∈X canonically determines a representation ofA.

In the sequel, we shall define completely regular and compact representations and, finally, we shall prove that any compact representation arises canonically from a filter space of the BL-algebra satisfying certain conditions.