CEJM 2(2003)221{237

Onthe lattice of deductive systems ofa BL-algebra

DumitruBu» sneag ¤, Dana Piciuy University ofCraiova, 1100 Craiova, Romania

Received 9January2003; revised 13March 2003

Abstract: Fora BL-algebra A we denote by Ds(A)the lattice ofall deductive systems of A.Theaim of this paperis toput in evidence newcharacterizations for the meet-irreducible elements on Ds(A). Hyperarchimedean BL-algebras, too, arecharacterized. c Central EuropeanScience Journals. All rights reserved. ® Keywords: BL-algebra, Booleanalgebra, Archimedean,Hyperarchime deanBL-algebra, Deductive system, Irreducible element, Prime deductive system, Maximal deductive system. MSC(2000):03G10

1Introduction

The origin of BL-algebras isinMathematical Logic;they where invented by H¶ajek in [8] inorder to study the ,,BasicLogic" (BL, for short) arising from the continuous triangular norms, familiar in the framework of fuzzyset theory.They playthe roleof Lindenbaum algebras from classicalPropositional calculus.Apart from their logicalinterest, BL- algebras haveinteresting algebraicproperties (see[9], [10], [12]). The paper isorganized as follows. Insection 2we recallthe basicde¯ nitions and put in evidencemany rules of calculus inBL-algebras which we need inthe rest of paper. Section 3contains someresults relativeto the latticeof deductive systems of aBL- algebra (Theorem 3.13characterizes the BL-algebras for which the latticeof deductive systems isa Boolean lattice). Section 4contains new characterizations for prime and completelymeet-irreducible deductive systems of aBL-algebra (seeProposition 4.9,Corollary 4.11,Theorem 4.12, Theorem 4.19,Theorem 4.20and Corollary 4.21).

¤ E-mail:[email protected] y E-mail:[email protected] 222 D.Bu¹sneag,D. Piciu/ CentralEuropean Journal of Mathematics2 (2003)221{237

In section 5we introduce the notions of archimedean and hyperarchimedean BL- algebra and we prove atheorem ofNachbin type for BL-algebras (seeTheorem 5.15). These results are inthe general spirit of algebras oflogic,as exposed in [11].

2De¯nitions and ¯rstproperties

De¯nition 2.1. A BL-algebra ([8]{[10],[12])is an algebra

(A; ; ; ; ; 0; 1) ^ _ ­ ! of type (2,2,2,2,0,0)satisfying the following: (a ) (A; ; ; 0; 1)isa bounded lattice, 1 ^ _ (a ) (A; ; 1)isa commutative monoid, 2 ­ (a ) and form an adjoint pair, i.e. c a b i® a c b for all a; b; c A (where 3 ­ ! µ ! ­ µ 2 isthe latticeordering on A), µ (a ) a b = a (a b), 4 ^ ­ ! (a ) (a b) (b a) = 1, for all a; b A. 5 ! _ ! 2 Examples (E )De¯ne onthe realunit interval I = [0; 1]binary operations and by 1 ­ ! x y = max 0; x + y 1 ­ f ¡ g x y = min 1; 1 x + y : ! f ¡ g Then (I; ; min; max; ; ; 0; 1) isa BL-algebra (called Lukasiewicz structure ). µ ­ ! (E2)De¯ne onthe realunit interval I

x y = min x; y ­ f g x y = 1 i® x y and y otherwise. ! µ Then (I; ; min; max; ; ; 0; 1) isa BL-algebra (called G}odelstructure ). µ ­ ! (E ) Let be the usual multiplication ofrealnumbers on the unit interval Iand 3 ­ x y = 1 i® x y and y=x otherwise. Then ( I; ; min; max; ; ; 0; 1) is a BL- ! µ µ ­ ! algebra (called productstructure or Gaines structure ).

Remark 2.2. Not everyresiduated lattice,however, is a BL-algebra (see[12], p.16). Consider, for examplea residuated latticede¯ ned on the unit interval,for all x; y; z I, 2 such that 1 x y = 0; i® x + y and x y elsewhere ­ µ 2 ^ 1 x y = 1 if x y and max x; y elsewhere. ! µ f2 ¡ g D.Bu¹sneag,D. Piciu/ CentralEuropean Journal of Mathematics2 (2003)221{237 223

Let 0 < y < x; x + y < 1 : Then y < 1 x and 0 = y = x y; but x (x y) = 2 2 ¡ 6 ^ ­ ! x ( 1 x) = 0: Therefore (a )does not hold. ­ 2 ¡ 4 (E ) If (A; ; ; ; 0; 1) is a Booleanalgebra , then (A; ; ; ; ; 0; 1) isa BL-algebra 4 ^ _ : ^ _ ­ ! where the operation coincideswith and x y = x y for all x; y A: ­ ^ ! : _ 2 (E ) If (A; ; ; ; 0; 1) is a relativeStone lattice (see[1], p.176), then 5 ^ _ ! (A; ; ; ; ; 0; 1)isaBL-algebra where the operation coincidewith . ^ _ ­ ! ­ ^ (E ) If (A; ;¤ ; 0) is a MV-algebra (see[2], [3], [12]), then ( A; ; ; ; ; 0; 1) is a BL- 6 © ^ _ ­ ! algebra, where for x; y A : 2

x y = (x¤ y¤)¤; ­ ©

x y = x¤ y; 1 = 0¤; ! ©

x y = (x y) y = (y x) x and x y = (x¤ y¤)¤: _ ! ! ! ! ^ _ ABL-algebra is nontrivial if 0 =1.F or any BL-algebra A,the reduct L(A) = 6 (A; ; ; 0; 1) isa bounded distributive lattice.F or any a A , we de¯ne a¤ = a 0 ^ _ 2 0 ! and denote (a¤)¤ by a¤¤: Wedenote the set of natural numbers by ! and de¯ne a = 1 n n 1 and a = a ¡ a for n ! 0 : The order of a A; a =1,in symbols ord(a) is the ­ 2 nf g 2 6 smallest n ! such that an =0;ifno such n exists,then ord(a) = : 2 1 ABL-algebra iscalled locally¯ nite if allnon unit elementsin it are of ¯nite order. In [8]-[10],[12] it isproved that if Aisa BL-algebra and a; b; c; b A, ( i I) then i 2 2 we havethe following rules of calculus: (c ) a b a; b; hence a b a b and a 0 = 0; 1 ­ µ ­ µ ^ ­ (c ) a b implies a c b c; 2 µ ­ µ ­ (c ) a b i® a b = 1; 3 µ ! (c ) 1 a = a; a a = 1; a b a; a 1 = 1; 4 ! ! µ ! ! (c ) a a¤ = 0; 5 ­ (c ) a b = 0 i® a b¤; 6 ­ µ (c ) a b = 1 implies a b = a b; 7 _ ­ ^ (c ) a (b c) = (a b) c = b (a c); 8 ! ! ­ ! ! ! (c ) (a b) (a c) = (a b) c; 9 ! ! ! ^ ! (c ) a (b c) (a b) (a c); 10 ! ! ¶ ! ! ! (c ) a b implies c a c b; b c a c and b¤ a¤; 11 µ ! µ ! ! µ ! µ (c ) a (a b) b , ((a b) b) b = a b; 12 µ ! ! ! ! ! ! (c ) a (b c) = (a b) (a c); 13 ­ _ ­ _ ­ (c ) a (b c) = (a b) (a c); 14 ­ ^ ­ ^ ­ (c ) a b = ((a b) b) ((b a) a); 15 _ ! ! ^ ! ! (c ) (a b)n = an bn; (a b)n = an bn; hence a b = 1 implies an bn = 1 for any 16 ^ ^ _ _ _ _ n !; 2 (c ) a (b c) = (a b) (a c); 17 ! ^ ! ^ ! (c ) (b c) a = (b a) (c a); 18 ^ ! ! _ ! (c ) (a b) c = (a c) (b c); 19 _ ! ! ^ ! 224 D.Bu¹sneag,D. Piciu/ CentralEuropean Journal of Mathematics2 (2003)221{237

(c ) a b (b c) (a c); 20 ! µ ! ! ! (c ) a b (c a) (c b); 21 ! µ ! ! ! (c ) a b (a c) (b c); 22 ! µ ­ ! ­ (c ) a (b c) b (a c); 23 ­ ! µ ! ­ (c ) (b c) (a b) a c; 24 ! ­ ! µ ! (c25) (a1 a2) (a2 a3) : : : (an 1 an) a1 an; ! ­ ! ­ ­ ¡ ! µ ! (c ) a; b c and c a = c b implies a = b; 26 µ ! ! (c ) a (b c) (a b) (a c); hence am bn (a b)mn, for any m; n !; 27 _ ­ ¶ _ ­ _ _ ¶ _ 2 (c ) (a b) (a0 b0) (a a0) (b b0); 28 ! ­ ! µ _ ! _ (c ) (a b) (a0 b0) (a a0) (b b0); 29 ! ­ ! µ ^ ! ^ (c ) (a b) c ((b a) c) c; 30 ! ! µ ! ! ! (c31)

a ( bi) (a bi); ­ i I µ i I ­ 2 2 V V a ( bi) = (a bi); ­ i I i I ­ 2 2 W W a ( bi) = (a bi); ! i I i I ! 2 2 V V ( bi) a = (bi a) i I ! i I ! 2 2 W V (bi a) ( bi) a; i I ! µ i I ! 2 2 W V (a bi) a ( bi); i I ! µ ! i I 2 2 W W a ( bi) = (a bi); if A isan BL-chain then a ( bi) = (a bi) ^ i I i I ^ _ i I i I _ 2 2 2 2 W W V V (whenever the arbitrary meetsand unions exist) ;

(c ) a a¤¤ , 1¤ = 0 , 0¤ = 1; a¤¤¤ = a; a¤¤ a¤ a; 32 µ µ ! (c ) (a b)¤ = a¤ b¤ and (a b)¤ = a¤ b¤; 33 ^ _ _ ^ (c ) (a b)¤¤ = a¤¤ b¤¤ , (a b)¤¤ = a¤¤ b¤¤; (a b)¤¤ = a¤¤ b¤¤ , (a b)¤¤ = a¤¤ b¤¤; 34 ^ ^ _ _ ­ ­ ! ! (c ) If a¤¤ a¤¤ a; then a¤¤ = a; 35 µ ! (c ) a = a¤¤ (a¤¤ a); 36 ­ ! (c ) a b¤ = b a¤ = a¤¤ b¤ = (a b)¤; 37 ! ! ! ­ (c ) (a¤¤ a)¤ = 0; (a¤¤ a) a¤¤ = 1; 38 ! ! _ (c ) b¤ a implies a (a b)¤¤ = b¤¤: 39 µ ! ­ For any BL-algebra A, B(A)denotes the Boolean algebra of allcomplemented ele- ments in L(A) (hence B(A) = B(L(A))):

Proposition 2.3. ([8]-[10],[12])F or e A; the following are equivalent: 2 (i) e B(A); 2 (ii) e e = e and e = e¤¤; ­ (iii) e e = e and e¤ e = e; ­ ! D.Bu¹sneag,D. Piciu/ CentralEuropean Journal of Mathematics2 (2003)221{237 225

(iv) e e¤ = 1: _

Remark 2.4. If a A; and e B(A); then e a = e a; a e = (a e¤)¤ = a¤ e; if 2 2 ­ ^ ! ­ _ e a a¤, then e a B(A): µ _ ­ 2 Proposition 2.5. For e A; the following are equivalent: 2 (i) e B(A); 2 (ii) (e x) e = e; for every x A: ! ! 2 Proof.

(i) (ii) If x A; then from 0 x we deduce e¤ e x hence ) 2 µ µ ! (e x) e e¤ e = e: Since e (e x) e we obtain (e x) e = e: ! ! µ ! µ ! ! ! ! (ii) (i) If x A; then from (e x) e = e we deduce ) 2 ! ! (e x) [(e x) e] = (e x) e; hence (e x) e = e x: For x = 0 we obtain ! ­ ! ! ! ­ ! ^ ^ that e¤ e = 0: Also,from hypothesis (for x =0)we obtain e¤ e = e: So, from (c ) ^ ! 15 we obtain

e e¤ = [(e e¤) e¤] [(e¤ e) e] _ ! ! ^ ! ! = [(e e¤) e¤] (e e) ! ! ^ ! = [(e e¤) e¤] 1 ! ! ^ = (e e¤) e¤ ! ! = [e (e e¤)]¤ (by (c )) ­ ! 37 = (e e¤)¤ = 0¤ = 1; ^ hence e B(A): 2 De¯nition 2.6. Following Diego[5], by Hilbert algebra we mean an algebra ( A; ; 1) of ! type (2; 0)satisfying the following identities: (a ) x (y x) = 1; 6 ! ! (a ) (x (y z)) ((x y) (x z)) = 1; 7 ! ! ! ! ! ! (a ) If x y = y x = 1, then x = y: 8 ! ! Proposition 2.7. For aBL-algebra ( A; ; ; ; ; 0; 1) the following are equivalent: ^ _ ­ ! (i) (A; ; 1) isa Hilbert algebra, ! (ii) (A; ; ; ; 0; 1) isarelativeStone lattice. ^ _ ! Proof. (i) (ii)Suppose that ( A; ; 1) isaHilbert algebra, then for every ) ! x; y; z A we have 2 x (y z) = (x y) (x z) ! ! ! ! !

From (c8) and (c9) we have

x (y z) = (x y) z ! ! ­ ! 226 D.Bu¹sneag,D. Piciu/ CentralEuropean Journal of Mathematics2 (2003)221{237

and

(x y) (x z) = (x y) z; ! ! ! ^ ! so we obtain

(x y) z = (x y) z ­ ! ^ ! hence x y = x y; that is (A; ; ; ; 0; 1) isa relativeStone lattice. ­ ^ ^ _ ! (ii) (i) If (A; ; ; ; 0; 1)isarelativeStone lattice,then ( A; ; ; ; 0; 1)isaHeyting ) ^ _ ! ^ _ ! algebra, so ( A; ; 1) isa Hilbert algebra. !

3Thelattice of deductive systemsof a BL-algebra

Inthe rest of this paper by Awe denote aBL-algebra.

De¯nition 3.1. Anon empty subset D A is a deductivesystem of A,ds for short, if ³ the following conditions are satis¯ed: (a ) 1 D; 6 2 (a ) If x; x y D, then y D: 7 ! 2 2 Clearly 1 and A are ds; ads of Aiscalled proper if D = A: f g 6

Remark 3.2. A ds D isproper i®0 = D i®no element a A holds a; a¤ D: 2 2 2 Remark 3.3. ([12])A non empty subset D A is a ds of A, i® for all a; b A : ³ 2 (a ) a; b D implies a b D; 8 2 ­ 2 (a ) a D and a b implies b D: 9 2 µ 2 Deductive systems of A and congruence relations on A: ¹ D x y i® (x y) (y x) D ¹ D ! ­ ! 2 are inone-to-one correspondence ([12],p.21). Starting from ads D,the quotient algebra A=D becomesa BL-algebra with the natural operations induced from those of A. We let x=D bethe congruence classof x modulo , x A: vD 2 Then, for x; y A, x=D y=D i® x y D and x=D = 1=D i® x D: 2 µ ! 2 2 Remark 3.4. Deductive systems are calledalso implicative¯ lters inliterature. Toavoid confusion we reserve,however, the name ¯lter to lattice¯ lters in this paper. From ( c1) and Remark3.3 we deduce that everyds of A isa ¯lter for L(A),but ¯lters of L(A) are not, in general,deductive systems for A;in example( E ); the onlyproper ds is 1 : 1 f g We denote by Ds(A)the set ofalldeductive systems ofA. D.Bu¹sneag,D. Piciu/ CentralEuropean Journal of Mathematics2 (2003)221{237 227

For anonempty subset M A we denote by [ M ) the ds of A generated by M (that ³ is, [M ) = D Ds(A) : M D ): \f 2 ³ g If M = a with a A; we denote by [ a)the ds generated by a ([a) is called f g 2 f g principal). For D Ds(A) and a A D; we denote by D(a) = [D a ): 2 2 n [ f g Proposition 3.5. ([8]-[10],[12]) (i) If M A isanonempty subset ofA,then: ³

[M ) = a A : x : : : x a; for some x :::; x M : f 2 1 ­ ­ n µ 1; n 2 g

Inparticular, for a A; 2

[a) = x A : x an; for some n ! : f 2 ¶ 2 g

(ii) If D Ds(A) and a A D; then 2 2 n

D(a) = x A : x y an; with y D and n ! : f 2 ¶ ­ 2 2 g

(iii) If x; y A; and x y, then [y) [x): 2 µ ³ (iv) If x; y A; then [x) [y) = [x y): 2 \ _ Remark 3.6. ([12],p.17) If D Ds(A) and a A; then a D; i® an D; for any 2 2 2 2 n !: 2 For D ; D Ds(A) we put 1 2 2

D D = D D and 1 ^ 2 1 \ 2

D D = [D D ) = a A : a x y; for some x D and y D : 1 _ 2 1 [ 2 f 2 ¶ ­ 2 1 2 2g

Then (Ds(A); ; ; 1 ; A) is a completeBrouwerian lattice ;we recallthat acomplete ^ _ f g latticeis Brouwerian ifitsatis¯es the identity a ( bi) = (a bi),whenever the arbitrary ^ i i ^ unions exists. W W

De¯nition 3.7. ([6],p.93) Let L be acompletelattice and let a be an elementof L. Then aiscalled compact if a X for some X L impliesthat a X for some¯ nite µ _ ³ µ _ 1 X X: Acompletelattice is called algebraic if everyelements is the join of compact 1 ³ elements(in the literature, algebraiclattices are alsocalled compactlygenerated lattices ).

Proposition 3.8. The lattice( Ds(A); )isanalgebraiclattice. ³ Proof. Weknow that ( Ds(A); )iscomplete. W eclaimthat for a A; [a) is a ³ 2 228 D.Bu¹sneag,D. Piciu/ CentralEuropean Journal of Mathematics2 (2003)221{237 compact elementof Ds(A): Let X Ds(A) and let [a) D: We have ³ ³ D X 2 W D = x A : x : : : x x for some x D ; D X : f 2 1 ­ ­ n µ i 2 i i 2 g D X _2 Therefore, a x : : : x , x D ; D X; 1 i n: Then, with ¶ 1 ­ ­ n i 2 i i 2 µ µ X1 = D1; :::; Dn , [a) D: Sincefor any D Ds(A) we have D = [a), we see f g ³ D X 2 a D 2 1 2 that Ds(A)isalgebraic. W W

Lemma 3.9. If x; y A; then [x) [y) = [x y): 2 _ ­ Proof. Since x y x; y; then [x); [y) [x y); hence [x) [y) [x y): ­ µ ³ ­ _ ³ ­ If z [x y); then for somenatural number n; z (x y)n = xn yn [x) [y); 2 ­ ¶ ­ ­ 2 _ hence z [x) [y); that is [x y) [x) [y); so [x) [y) = [x y): 2 _ ­ ³ _ _ ­ For D ; D Ds(A) we put D D = a A : D [a) D : 1 2 2 1 ! 2 f 2 1 \ 2 2g Lemma 3.10. If D ; D Ds(A) then 1 2 2 (i) D D Ds(A); 1 ! 2 2 (ii) If D Ds(A); then D D D i® D D D (that is, 2 1 \ ³ 2 ³ 1 ! 2 D D = sup D Ds(A) : D D D ): 1 ! 2 f 2 1 \ ³ 2g Proof. (i) Since [1) = 1 and [1) D = 1 D we deduce that 1 D D . f g \ 1 f g ³ 2 2 1 ! 2 Let x; y A such that x y and x D D , that is [x) D D : Then 2 µ 2 1 ! 2 \ 1 ³ 2 [y) [x); so [y) D [x) D D ; hence [y) D D that is y D D . ³ \ 1 ³ \ 1 ³ 2 \ 1 ³ 2 2 1 ! 2 Toprove that ( a )isveri¯ ed, let x; y A such that x; y D D , hence 8 2 2 1 ! 2 [x) D D and [y) D D : \ 1 ³ 2 \ 1 ³ 2 We deduce ([x) D ) ( [y) D ) D ; hence ([x) [y)) D D : By Lemma 3.9 \ 1 _ \ 1 ³ 2 _ \ 1 ³ 2 we deduce that [ x y) D D ; that is x y D D ; hence D D Ds(A) ­ \ 1 ³ 2 ­ 2 1 ! 2 1 ! 2 2 (by Remark3.3) : (ii) Suppose D D D and let x D: Then [x) D; hence [x) D D D D : 1 \ ³ 2 2 ³ \ 1 ³ \ 1 ³ 2 So x D D , that is D D D : 2 1 ! 2 ³ 1 ! 2 Suppose D D D and let x D D. Then x D, hence x D D , that ³ 1 ! 2 2 1 \ 2 2 1 ! 2 is [x) D D : Since x [x) D D we obtain x D ; that is D D D . \ 1 ³ 2 2 \ 1 ³ 2 2 2 1 \ ³ 2 Remark3.11. From Lemma3.10 we deduce that ( Ds(A); ; ; ; 1 ; A) is a Heyting _ ^ ! f g algebra; for D Ds(A), D¤ = D 0 = D 1 = x A : [x) D = 1 and so, for 2 ! ! f g f 2 \ f gg a A, [a)¤ = x A : [x) [a) = 1 = x A : [x a) = 1 = x A : x a = 1 : 2 f 2 \ f gg f 2 _ f gg f 2 _ g

Proposition 3.12. If x; y A; then [x y)¤ = [x)¤ [y)¤. 2 ­ \

Proof. If a [x y)¤, then a (x y) = 1: Since x y x; y, then a x = a y = 1; 2 ­ _ ­ ­ µ _ _ D.Bu¹sneag,D. Piciu/ CentralEuropean Journal of Mathematics2 (2003)221{237 229

hence a [x)¤ [y)¤; that is [x y)¤ [x)¤ [y)¤: 2 \ ­ ³ \ Let now a [x)¤ [y)¤; that is a x = a y = 1: 2 \ _ _ By (c ) we deduce a (x y) (a x) (a y) = 1; hence a (x y) = 1; that is 27 _ ­ ¶ _ ­ _ _ ­ a [x y)¤. 2 ­ Itfollows that [ x)¤ [y)¤ [x y)¤; hence [x y)¤ = [x)¤ [y)¤. \ ³ ­ ­ \ Theorem3.13. If A isa BL-algebra, then the following assertions are equivalent: (i) (Ds(A); ; ;¤ ; 1 ; A)isa Boolean algebra, _ ^ f g (ii) Every ds of A isprincipal and for every x A;there is n ! such that n 2 2 x (x )¤ = 1: _ Proof.

(i) (ii) Let D Ds(A) ; since Ds(A)isBoolean algebra, then D D¤ = A: So, for ) 2 _ 0 A; there exist a D; b D¤ such that a b = 0: 2 2 2 ­ Since b D¤,by Remark3.11, it follows that a b = 1: By (c )we deduce that 2 _ 7 a b = a b = 0; that is b isthe complement of ain L(A): Hence a; b B(A) = ^ ­ 2 B(L(A)): If x D; since b D¤ we have b x = 1: Since a = a (b x) = (a b) (a x) = a x 2 2 _ ^ _ ^ _ ^ ^ we deduce that a x; that is D = [a): Henceevery ds of A isprincipal. µ Let now x A; since Ds(A)isBoolean algebra, then [ x) [x)¤ = A [x)¤(x) = A 2 n _ , , a A : a c x ; with c [x)¤ and n ! = A (seeProposition 3.5,(ii)). f 2 ¶ ­ 2 2 g n So, since 0 A; there exist c [x)¤ and n ! such that c x = 0: Since c [x)¤, 2 2 n 2 n ­ 2 n then x c = 1: By (c6), from c x = 0 we deduce c (x )¤: So, 1 = x c x (x )¤; _ n ­ µ _ µ _ hence x (x )¤ = 1: _ (ii) (i)ByRemark 3.11, Ds(A)isa Heyting algebra. Toprove Ds(A) is Boolean ) algebra, we must show that for D Ds(A), D¤ = 1 only for D = A ([1],p. 175). 2 f g Byhypothesis everyds of A isprincipal, so we have a A such that D = [a): 2 n Also,by hypothesis, for a A, there is n ! such that a (a )¤ = 1: By Remark n 2 n 2 n _ 3.11, (a )¤ [a)¤ = 1 ; hence (a )¤ = 1; that is a = 0: ByRemark 3.6, we deduce 2 f g that 0 D, hence D = A: 2

4Thesp ectrum ofa BL-algebra

De¯nition 4.1. Let L be alatticewith the leastelement 0 and the greatest element1. An element p < 1 is meet-irreducible if p = x y implies p = x or p = y; an element ^ p < 1 is meet-prime if x y p implies x p or y p: Dually are de¯ned the notions ^ µ µ µ of join-irreducible and join-prime.

Remark 4.2. If L isdistributive meet-irreducible and meet-prime elementsare the same.

These de¯nitions can be extended to arbitrary meetsand we obtain the concepts of completelymeet (join)-irreducible and completelymeet (join)-prime elements,which are 230 D.Bu¹sneag,D. Piciu/ CentralEuropean Journal of Mathematics2 (2003)221{237 no longer equivalent. For the lattice Ds(A)(which isdistributive) we denote by Spec(A)the set of all meet-irreducible (hence meet-prime) elements( Spec(A)iscalledthe spectrum of A) and by Irc(A)the set ofallcompletely meet-irreducible elementsof the lattice Ds(A):

De¯nition 4.3. ([12],p.18) Aproper ds D of A is called prime if, for any a; b A; the 2 condition a b D implies a D or b D: _ 2 2 2 Remark 4.4. ([12],p.18,19) 1: Anon-degenerate BL-algebra contains aprime ds. 2: If D isa prime ds of A then, for any a; b A; either a b D or b a D: 2 ! 2 ! 2 Moreover, A islinear i®any proper ds of A is prime. 3: If P isa prime ds of A and D isa proper ds of A such that P D; then also D is ³ prime.

Theorem4.5. ([10],[12])F or aproper P Ds(A)the following are equivalent: 2 (i) P is prime, (ii) A=P is a chain, (iii) For all x; y A; x y P or y x P; 2 ! 2 ! 2 (iv)The set of proper ds including aprime ¯lter P of A is a chain.

Theorem4.6. (Primeds theorem [10]) If D Ds(A) and I an idealof the lattice L(A) 2 such that D I = ?; then there isa prime ds P of A such that D P and P I = ?: \ ³ \ Corollary4.7. If D Ds(A)isproper and a A D; then there is P Spec(A) such 2 2 n 2 that D P and a = P: In particular, for D = 1 we deduce that for any a A; a = 1; ³ 2 f g 2 6 there is P Spec(A); such that a = P : a 2 2 a Proposition 4.8. For a proper P Ds(A)the following are equivalent: 2 (i) P is prime, (ii) P Spec(A); 2 (iii) If a; b A; and a b = 1; then a P or b P: 2 _ 2 2 Proof. (i) (ii) Let D ; D Ds(A) such that D D = P: ) 1 2 2 1 \ 2 Since P D ; P D ; by Theorem 4.5,(iv), D D or D D ; hence P = D ³ 1 ³ 2 1 ³ 2 2 ³ 1 1 or P = D2: (ii) (i) Let a; b A; such that a b P: ) 2 _ 2 Since P (a) P (b) = (P [a)) (P [b)) = P ([a) [b)) = P [a b) = P; then \ _ \ _ _ \ _ _ P = P (a) or P = P (b), hence a P or b P; that is P is prime. 2 2 (i) (iii)Clearly,since1 P: ) 2 (iii) (i)Clearlyby Theorem 4.5,(iii) (since ( a b) (b a)=1for every a; b A). ) ! _ ! 2 D.Bu¹sneag,D. Piciu/ CentralEuropean Journal of Mathematics2 (2003)221{237 231

Proposition 4.9. For a proper P Ds(A)the following are equivalent: 2 (i) P Spec(A); 2 (ii) For every x; y A P there is z A P such that x z and y z: 2 n 2 n µ µ Proof. (i) (ii) Let P Spec(A) and x; y A P .Ifby contrary,for every a A with x a ) 2 2 n 2 µ and y a then a P; since x; y x y we deduce x y P: Hence, x P or y P; µ 2 µ _ _ 2 2 2 acontradiction. (ii) (i)Isuppose by contrary that there exist D ; D Ds(A) such that ) 1 2 2 D D = P; and P = D ; P = D : So, we have x D P and y D P: By 1 \ 2 6 1 6 2 2 1n 2 2n hypothesis there is z A P such that x z and y z: 2 n µ µ We deduce z D D = P -acontradiction. 2 1 \ 2 Corollary4.10. For a proper P Ds(A)the following are equivalent: 2 (i) P Spec(A); 2 (ii) If x; y A and [x) [y) P; then x P or y P: 2 \ ³ 2 2 Proof. (i) (ii) Let x; y A such that [x) [y) P and suppose by contrary that x; y = ) 2 \ ³ 2 P: Then by Proposition 4.9there is z A P such that x z and y z: Hence 2 n µ µ z [x) [y) P; so z P ,acontradiction. 2 \ ³ 2 (ii) (i) Let x; y A such that x y P: Then [x y ) P . ) 2 _ 2 _ ³ Since [x y) = [x) [y)(by Proposition 3.5,(iv)) we deduce that [ x) [y) P; hence _ \ \ ³ x P or y P; that is P Spec(A). 2 2 2 Corollary4.11. For a proper P Ds(A)the following are equivalent: 2 (i) P Spec(A); 2 (ii) For every x; y A=P; x = 1; y = 1 there is z A=P; z = 1 such that x z, y z: 2 6 6 2 6 µ µ Proof. (i) (ii)Clearly,by Proposition 4.9,since if x = a=P; with a A; then the condition ) 2 x = 1 (in A=P )isequivalentwith a = P: 6 2 (ii) (i) Let x; y A=P: Then x = a=P = 1 and y = b=P = 1 (in A=P ).Byhypothesis ) 2 6 6 there is z = c=P = 1 (that is c = P ) such that x; y z equivalentwith a c; b 6 2 µ ! ! c P: If consider d = (b c) ((a c) c) then by (c ) and (c )we deduce that 2 ! ! ! ! 4 8 a; b d: Clearly d = P ,henceby Proposition 4.9we deduce that P Spec(A): µ 2 2 Theorem4.12. For a proper P Ds(A)the following are equivalent: 2 (i) P Spec(A); 2 (ii) For every D Ds(A); D P = P or D P: 2 ! ³ Proof. (i) (ii) Let P Spec(A): Since Ds(A)isa Heyting algebra (by Remark3.11) for ) 2 232 D.Bu¹sneag,D. Piciu/ CentralEuropean Journal of Mathematics2 (2003)221{237

D Ds(A) we have P = (D P ) ((D P ) P ) and so P = D P or 2 ! \ ! ! ! P = (D P ) P: If P = (D P ) P then D P: ! ! ! ! ³ (ii) (i) Let D ; D Ds(A) such that D D = P: Then D D P (see Lemma ) 1 2 2 1 \ 2 1 ³ 2 ! 3.10,(ii)) and so,if D P; then P = D and if D P = P; then P = D ; hence 2 ³ 2 2 ! 1 P Spec(A): 2 Werecallthat if L is a pseudocompleteddistributive lattice ,then two subsets associated with L ([1],p.153) are Rg(L) = x L : x¤¤ = x and D(L) = x L : x¤ = 0 . f 2 g f 2 g The elementsof Rg(L) are called regular and those of D(L) dense. Note that 0; 1 Rg(L); 1 D(L) and D(L)isa ¯lter in L and Rg(L)isa Boolean algebra under f g ³ 2 the operations induced by the ordering on L ([1],p.157) .

Corollary4.13. For aBL-algebra A; Spec(A) D(Ds(A)) Rg(Ds(A)): ³ [

Proof. Let P Spec(A) and D = P ¤ Ds(A);then by Theorem 4.12, D P or 2 2 ³ D P = P equivalentwith P ¤ P or P ¤ P = P: Since Ds(A)isa Heyting algebra ! ³ ! then P ¤ P = P ¤¤; so P ¤¤ = A or P ¤¤ = P equivalentwith P ¤ = 1 or P ¤¤ = P; that ! f g is P D(Ds(A)) Rg(Ds(A)): 2 [ Remark4.14. From Corollary 4.7we deduce that for every D Ds(A), 2

D = P Spec(A) : D P and P Spec(A) = 1 : \f 2 ³ g \ f 2 g f g Relativeto the uniqueness of deductive systems asintersection of primes we have:

Theorem4.15. If every D Ds(A)has aunique representation as an intersection of 2 elements of Spec(A); then (Ds(A); ; ;¤ ; 1 ; A isa Boolean algebra. _ ^ f g g

Proof. Let D Ds(A) and D0 = M Spec(A) : D * M Ds(A): By Remark 2 \f 2 g 2 4.14, D D0 = M Spec(A) = 1 ; if D D0 = A; then by Corollary 4.7there \ \f 2 g f g _ 6 exists D00 Spec(A) such that D D0 D00 and D00 = A: Consequently, D0 has two 2 _ ³ 6 representations D0 = M Spec(A) : D * M = D00 ( M Spec(A) : D * M ); \f 2 g \ \f 2 g which iscontradictory.Therefore D D0 = A and so Ds(A)isa Boolean lattice. _ Lemma 4.16. If D Ds(A); D = A and a = D; then there exists D Ds(A) maximal 2 6 2 a 2 with the property that D D and a = D : ³ a 2 a

Proof. Let F = D0 Ds(A) : D D0 and a = D0 ; clearly D F : D;a f 2 ³ 2 g 2 D;a If C is a chain in F then C F : ByZorn’s lemmathere existsa ds D which D;a [ 2 D;a a ismaximalsub ject to containing D and a = D : 2 a De¯nition 4.17. D Ds(A); D = A is called maximalrelative to a if a = D and if 2 6 2 D0 Ds(A)isproper such that a = D0, and D D0; then D = D0: 2 2 ³ D.Bu¹sneag,D. Piciu/ CentralEuropean Journal of Mathematics2 (2003)221{237 233

IfinLemma 4.16we consider D = 1 we obtain: f g Corollary4.18. For any a A; a = 1; there is a ds D maximalwith the property that 2 6 a a = D . 2 a Theorem4.19. For D Ds(A); D = A the following are equivalent: 2 6 (i) D Irc(A); 2 (ii) There is a A such that D ismaximal relative to a. 2 Proof. (i) (ii)See([7], p.248) (sinceby Proposition 3.8, Ds(A)isan algebraiclattice). ) (ii) (i) Let D Ds(A)maximalrelative to aand suppose D = Di with Di Ds(A) ) 2 i I 2 2 for every i I: Since a = D there is j I such that a = D : So,Ta = D and D D : 2 2 2 2 j 2 j ³ j Bythe maximalityof D we deduce that D = D ; that is D Irc(A): j 2 Theorem4.20. Let D Ds(A); D = A and a A D: Then the following are equivalent: 2 6 2 n (i) D ismaximalrelative to a; (ii) For every x A D there is n ! such that xn a D: 2 n 2 ! 2 Proof. (i) (ii) Let x A D: If a = D(x) = D [x); since D D(x) then D(x) = A (by the ) 2 n 2 _ » maximalityof D) hence a D(x)-acontradiction. Wededuce that a D(x); hence 2 2 a d xn; with d D and n !: Then d xn a, hence xn a D: ¶ ­ 2 2 µ ! ! 2 (ii) (i)Isuppose by contrary that there is D0 Ds(A); D0 = A such that a = D0 ) 2 6 2 and D D0: Then there is x0 D0 such that x0 = D; henceby hypothesis there is » n 2 n 2 n n ! such that x a D D0: Thus x a D0 and x D0; hence a D0 -a 2 0 ! 2 » 0 ! 2 0 2 2 contradiction.

Corollary4.21. For D Ds(A); D = A the following are equivalent: 2 6 (i) D Irc(A); 2 (ii) In the set A=D 1 we havean element p =1with the property that for every nf g 6 x A=D 1 there is n ! such that xn p: 2 nf g 2 µ Proof. (i) (ii)ByTheorem 4.19, D ismaximal relative to an element a = D;then, if we let ) 2 p = a=D A=D; p = 1 (since a = D)and for every x = b=D; x = 1 (that is b = D) by 2 6 2 6 2 Theorem 4.20there is n ! such that bn a D; that is xn p: 2 ! 2 µ (ii) (i) Let p = a=D A=D 1 ; (that is a = D) and x = b=D A=D 1 ; (that is ) 2 nf g 2 2 nf g b = D).Byhypothesis there is n ! such that xn p equivalentwith bn a D: 2 2 µ ! 2 Then by Theorem 4.20,we deduce that D Irc(A): 2 De¯nition 4.22. A ds of A is a minimalprime ds if P Spec(A)and, whenever 2 Q Spec(A) and Q P we have P = Q: 2 ³ 234 D.Bu¹sneag,D. Piciu/ CentralEuropean Journal of Mathematics2 (2003)221{237

Proposition 4.23. If P isa minimalprime ds, then for any a P there is b A P 2 2 n such that a b = 1: _ Proof. Let P beaminimalprime ds and a P: 2 Wede¯ne the set S = x A : there is b A P such that a b x : f 2 2 n _ ¶ g If b A P then a b b; so A P S: Moreover, a S because a 0 = a a and 2 n _ ¶ n ³ 2 _ ¶ 0 A P: 2 n Weshall prove that S isan idealof the lattice L(A). Let x; y A such that y S and x y: Thus, there is b A P such that a b 2 2 µ 2 n _ ¶ y x; hence a b x; so x S: ¶ _ ¶ 2 If x; y S then there are b; c A P such that a b x and a c y: If we suppose 2 2 n _ ¶ _ ¶ that b c P we get b P or c P because P isa prime ds. Thus, b c A P and _ 2 2 2 _ 2 n a (b c) x y; so x y S; hence S is an ideal. _ _ ¶ _ _ 2 Now, we suppose that 1 = S: Itfollows that 1 S = ? so,by Theorem 4.6,there is 2 f g \ a prime ds Q such that S Q = ?: Since A P S; we get Q P: But Q is prime and \ n ³ ³ P isminimal prime, so P = Q: On the other hand, a S; so a = Q: We get a P Q, 2 2 2 n which contradicts the fact that P = Q: Thus, our assumption that 1 = S is false. We 2 conclude that 1 S and our proof is¯nished. 2

5Maximaldeductive systems;archimedean and hyperarchimedeanBL-algebras

De¯nition 5.1. A ds of A is maximal ifitisproper and it isnot contained in any other proper ds.

Weshall denote by M ax(A)the set of allthe maximalds of A;it isobvious that M ax(A) Spec(A): ³ We have:

Theorem5.2. ([12],p.24) For M Ds(A); M = A; the following are equivalent: 2 6 (i) M M ax(A); 2 n (ii) For every x = M there is n ! such that (x )¤ M; 2 2 2 (iii) A=M islocally ¯ nite.

De¯nition 5.3. If D isa proper ds of A and there existsanother proper ds D0 such that

D D0 we say that D canbe extended to D0: ³ Theorem5.4. ([12],p.19) (i)Any proper ds D can beextended to aprime ds, (ii)Any proper ds D can beextended to amaximal,prime ds.

De¯nition 5.5. The intersection of the maximalds of A iscalled the radical of A. It D.Bu¹sneag,D. Piciu/ CentralEuropean Journal of Mathematics2 (2003)221{237 235 willbe denoted by Rad(A).Itisobvious that Rad(A) is a ds.

n Proposition 5.6. ([10]) Rad(A) = a A : (a )¤ a; for any n ! : f 2 µ 2 g

Proposition 5.7. For any a; b Rad(A); a¤ b¤ = 0: 2 ­

Proof. Let a; b Rad(A); to prove a¤ b¤ =0isequivalent with ( a¤ b¤)¤ = 1: Suppose 2 ­ ­ that (a¤ b¤)¤ = 1: ByCorollary 4.7,there isaprime ds P such that ­ 6 (a¤ b¤)¤ = P . By (c ) we have (a¤ b¤)¤ = a¤ b¤¤ = P; so by Theorem 4.5, ­ 2 37 ­ ! 2 b¤¤ a¤ P; that is (b¤¤ a)¤ P: ! 2 ­ 2 ByTheorem 5.4there isa maximalds M such that P M: Then b¤¤ a = M: By n n³ n ­ 2 Theorem 5.2,there is n ! such that [(b¤¤ a) ]¤ = [(b )¤¤ a ]¤ M ;so,if we let n n 2 ­ ­ 2 c = (b )¤¤ a ; we have c¤ M . Since a; b Rad(A)then we infer that a; b M; hence n ­ n 2 2 2 c = (b )¤¤ a M . Hence c and c¤ are in M which contradicts the fact that M is a ­ 2 proper ds of A.

n De¯nition 5.8. An element a of A is called in¯nitesimal if a = 1 and a a¤; for any 6 ¶ n !: 2 Proposition 5.9. For everynonunit element a of A the following are equivalent: (i) a isin¯nitesimal, (ii) a Rad(A): 2 Proof. (i) (ii) Let a =1be an in¯nitesimal and suppose a = Rad(A): Thus, there isa ) 6 2 maximal ds M of A such that a = M: ByTheorem 5.2,there is n ! such that n n 2 n 2 n (a )¤ M .Byhypothesis a a¤ hence (a )¤ a¤¤; so a¤¤ M; hence (a¤¤) = n 2 n ¶ µ 2 (a )¤¤ M: If we let b = (a )¤ we conclude that b; b¤ M which contradicts the fact 2 2 that M isaproper ds. n (ii) (i) Let a Rad(A); then (a )¤ a for any n !: For n =1we obtain that a¤ a: ) 2 n µ 2 n n n µ Sincefor any n !; a Rad(A)we deduce that ( a )¤ a : Since a¤ a a¤ a = 0 2 n 2 µ n ­ µ ­ we obtain that a¤ a = 0 for any n !; hence by (c6), a¤ (a )¤: So, for any n !; n n­ n 2 n µ 2 a¤ (a )¤ and (a )¤ a ; hence a¤ a ; that is a isan in¯nitesimal. µ µ µ

n n Lemma 5.10. If a A; n ! such that a (a )¤ = 1 and a a¤; then a = 1: 2 2 _ ¶

n n Proof. By (c11) we obtain (a )¤ a¤¤; so 1 = a (a )¤ a a¤¤ = a¤¤; hence a¤¤ = 1; µ _ µ _ 2 that is a¤ = 0: Then a (a 0) = a 0 = 0: From (c8)we deduce that ( a )¤ = 0: ! n! ! Recursivelywe obtain that ( a )¤ = 0: Then a 0 = 1; hence a = 1: _ Lemma 5.11. In any BL-algebra A the following are equivalent: n (i) For every a A; a a¤ for any n ! implies a = 1; 2 ¶ 2 (ii) For every a; b A; an b for any n ! implies a b = b and b a = a: 2 ¶ 2 ! ! 236 D.Bu¹sneag,D. Piciu/ CentralEuropean Journal of Mathematics2 (2003)221{237

Proof. n (i) (ii) Let a; b A such that a b for any n !: We get (a b)¤ = a¤ b¤ a¤ )n n 2 n ¶ 2 _ ^ µ µ a (a b) ; hence (a b) (a b)¤ for any n !.Byhypothesis, a b = 1: From µ _ _ ¶ _ 2 _ (c15) we deduce (a b) b = (b a) a = 1; hence a b = b and b a = a: ! ! n ! ! ! ! (ii) (i) Let a A such that a a¤ for any n !: Byhypothesis we get a a = a; ) 2 ¶ 2 ! so a = 1:

De¯nition 5.12. ABL-algebra A is called archimedean ifthe equivalentconditions from Lemma 5.11are satis¯ed.

Onecan easilyremark that aBL-algebra isarchimedean i®it has no in¯nitesimals.

De¯nition 5.13. Let A be aBL-algebra. An element a A is called archimedean if it 2 satis¯es the condition:

n there is n !; n 1; such that a (a )¤ = 1: 2 ¶ _ ABL-algebra A is called hyperarchimedean ifallits elementsare archimedean.

From Lemma 5.10we deduce:

Corollary5.14. Everyhyperarchimedean BL-algebra isarchimedean.

Now, we havea theorem of Nachbintype (see[1], p.73) for BL-algebras:

Theorem5.15. For aBL-algebra A the following are equivalent: (i) A ishyperarchimedean, (ii) For any ds D,the quotient BL-algebra A=D isan archimedean BL-algebra, (iii) Spec(A) = M ax(A); (iv)Any prime ds isminimal prime.

Proof. n (i) (ii) To prove A=D isarchimedean, let x = a=D A=D such that x x¤ for any ) 2 m ¶ n !: Byhypothesis, there is m !; m 1 such that a (a )¤ = 1: It follows that 2 m 2 ¶ m _ x (x )¤ = 1 (in A=D).In particular we have x x¤; so by Lemma 5.10we deduce _ ¶ that x = 1; that is A=D isarchimedean. (ii) (iii) Since M ax(A) Spec(A); we onlyhave to prove that any prime ds of A is ) ³ maximal. If P Spec(A); then A=P isa chain (seeTheorem 4.5).By hypothesis A=P 2 isarchimedean. ByTheorem 5.2to prove P M ax(A)it su±ces to prove that A=P 2 islocally¯ nite. n Let x = a=P A=P; x = 1: Then there is n !; n 1; such that x ¤ x¤: Since 2 n 6 n+1 2 ¶ n+1 A=P ischain we have x x¤. Thus x x x¤ = 0; hence x = 0; that is µ µ ­ o(x) < : It follows that A=P islocally¯ nite. 1 (iii) (iv) Let P; Q prime dssuch that P Q: Byhypothesis, P ismaximal, so P = Q: ) ³ D.Bu¹sneag,D. Piciu/ CentralEuropean Journal of Mathematics2 (2003)221{237 237

Thus Q isminimal prime. (iv) (i) Let a be anonunit elementfrom A.Weshall prove that a isanarchimedean ) element.If we denote D = [a)¤ = x A : a x = 1 (by Remark3.11), then f 2 _ g D Ds(A). Since a = 1; then a = D and we consider 2 6 2n D0 = D(a) = x A : x d a for some d D and n ! : Ifwe suppose that D0 f 2 ¶ ­ 2 2 g isaproper ds of A,then by Corollary 4.7,there isaprime ds P such that D0 P: By ³ hypothesis, P isa minimalprime. Since a P; using Proposition 4.23,we infer that 2 there is x A P such that a x = 1: It follows that x D D0 P; hence x P; 2 n _ 2 ³ ³ 2 so we get acontradiction. n Thus D0 isnot proper, so 0 D0; hencethere is n ! and d D such that d a = 0: n 2 n 2 2 ­ Thus d (a )¤ (by (c6)): We get a d a (a )¤: But a d = 1 (since d D), so we µ n _ µ _ _ 2 obtain that a (a )¤ = 1; that is a isan archimedean element. _

Acknowledgements

Wewould liketo express our gratitude for the guidance givenby the refereein the elaboration of this paper.

References

[1]R. Balbes and Ph. Dwinger: Distributive Lattices ,Universityof Missouri Press, 1974. [2]D. Bu»sneag and D.Piciu:\Meet-irreducible idealsin an MV-algebra", Analele Universit¸at»ii dinCraiova, Seria Matematica-Informatica ,Vol.XXVIII, (2001), pp. 110{119. [3]D. Bu»sneag and D.Piciu:\On the latticeof idealsof an MV-algebra", Scientiae MathematicaeJaponicae ,Vol.56, (2002), pp. 367{372. [4]R. Cignoli,I.M.L. D’Ottaviano, D.Mundici: Algebraicfoundation of many-valued reasoning,Kluwer AcademicPubl., Dordrecht, 2000. [5]A. Diego:\Sur lesalgµ ebres de Hilbert", In.Ed. Hermann: Collectionde Logique Math¶ematique ,SerieA, XXI,Paris, 1966. [6] G. GrÄatzer: Latticetheory ,W.H.Freeman and Company,San Francisco,1979. [7]G. Georgescu and M.Plo·s·cica:\V aluesand minimalspectrum of an algebraic lattice", Math.Slovaca ,Vol.52, (2002), pp. 247{253. [8]P .H¶ajek: Metamathematicsof Fuzzy Logic ,Kluwer AcademicPubl., Dordrecht, 1998. [9]A. Iorgulescu:\Is¶ ekialgebras. Connections with BL-algebras", to appear in Soft Computing. [10]A. Di Nola,G. Georgescu, A.Iorgulescu:\Pseudo-BL-algebras", to appear in MultipleV aluedLogic . [11]H. Rasiowa, An AlgebraicApproach to Non-Classical Logics ,PWNand North- Holland Publishing Company,1974. [12]E. Turunen: MathematicsBehind F uzzy Logic ,Physica-Verlag,1999.