Logic, Algebra and Truth Degrees 2008 September 8 to 11, Siena , Italy

1. On one-generated projective BL-algebrasAntonio Di Nola and Revaz GrigoliaUniversity of Salerno Tbilisi State UniversityDepartment of Mathematics Institute of Cybernetics and Informatics . Logic, Algebra and Truth Degrees2008 September 8 to 11, Siena , Italy

2. On one-generated projective BL-algebras LATD08Antonio Di Nola and Revaz Grigolia • BL-algebras are introduced by P. Hajek in [Metamathematics of fuzzy logic, Kluwer Academic Publishers, Dordrecht, 1998.] as an algebraic counterpart of the basic fuzzy propositional logic BL.

3. On one-generated projective BL-algebras LATD08Antonio Di Nola and Revaz Grigolia Basic fuzzy propositional logic is the logic of continuous t-norms. Formulas are built from propositional variables using connectives & (conjunction), → (implication) and truth constant 0 (denoting falsity). Negation ¬φ is defined as φ→0. Given a continuous t-norm * (and hence its residuum ) each evaluation e of propositional variables by truth degrees for [0,1] extends uniquely to the evaluation e*(φ) of each formula φ using * and  as truth functions of & and →. A formula φ is a t-tautology or standard BL-tautology if e*(φ) = 1 for each evaluation e and each continuous t-norm *.

4. On one-generated projective BL-algebras LATD08Antonio Di Nola and Revaz Grigolia • The following t-tautologies are taken as axioms of the logic BL: (A1) (φ → ψ) → ((ψ → χ) → (φ → χ)) (A2) (φ & ψ) → φ (A3) (φ & ψ) → (ψ & φ) (A4) (φ & (φ → ψ)) → (ψ & (ψ → φ)) (A5a) (φ → (ψ → χ)) → ((φ & ψ) → χ) (A5b) ((φ & ψ) → χ) → (φ → (ψ → χ)) (A6) ((φ → ψ) → χ) → (((ψ → φ) → χ) → χ) (A7) 0 → φ Modus ponens is the only inference rule: φ, φ → ψψ

5. On one-generated projective BL-algebras LATD08Antonio Di Nola and Revaz Grigolia BL-algebra(B; , , , ,0, 1)is a universal algebra of type(2,2,2,1,0,0)such that: • 1) (B; , ,0, 1) is a bounded distributive lattice; • 2) (B; , 1)is a commutativemonoid with identity: x  y = y  x, x (y  z) = (x  y)  z, x  1 = 1  x; • 3) (1) x (y  (x  y)) = x, (2) ((x  y)  x)  y = y, (3) (x  (x  y)) = 1, (4) ((x  z)  (z  (x _ y))) = 1, (5) (x  y)  z = (x  z)  (y  z), (6) x  y = x  (x  y), (7) x  y = ((x  y)  y)  ((y  x)  x), (8)x  y)  (y  x) = 1.

6. On one-generated projective BL-algebras LATD08Antonio Di Nola and Revaz Grigolia • BL-algebra B is named BL-chain if for every elements x, y B either x  y or y  x, where  is lattice order on B. • Let B1, B2 be BL-algebras, where B1 is BL-chain. Taking isomorphiccopies of the ones assume that 1B1= 0B2and (B1\ {1B1})  (B2\ {0B2}) = . • Let B1●B2 be the structure whose universe is B1B2 and x  y if (x, y B1and x 1 y) or (x, y B2 and x 2 y), or x B1and y B2 . Moreover, • x  y = x i y for x, y Bi, x  y = x for x B1and y B2 ; • x  y = 1B2 for x  y; • for x > y we put x  y = x i y if x, y Bi and • putx  y = y for x B2 and y B1\ B2.

7. On one-generated projective BL-algebras LATD08Antonio Di Nola and Revaz Grigolia According to the definition we have B1 B2 B1●B2

8. On one-generated projective BL-algebras LATD08Antonio Di Nola and Revaz Grigolia Proposition 1.B = B1●B2is aBL-algebra with 0B = 0B1 ; 1B = 1B2 and 1B1 = 0B2being non-extremal idempotent. Moreover, ifB1,B2areBL-chains, then B =B1●B2is BL-chain too.

9. On one-generated projective BL-algebras LATD08Antonio Di Nola and Revaz Grigolia • The variety BL of all BL-algebras is not locally finite and it is generatedby all finite BL-chains. • In addition, we have that the subvarieties of BL,which are generated by finite families of finite BL-chains, are locally finite.

10. On one-generated projective BL-algebras LATD08Antonio Di Nola and Revaz Grigolia • A BL-algebra is said to be an MV -algebra, if it satises the followingequation: x = x, where x = x  0. More precisely,

11. On one-generated projective BL-algebras LATD08Antonio Di Nola and Revaz Grigolia • An algebra A = (A;,,, 0,1), is said to be anMV-algebra, if it satises the following equations: (i) (x  y)  z = x  (y  z); (ii) x  y = y  x; (iii) x  0 = x; (iv) x  1 = 1; (v) 0 = 1; (vi) 1 = 0; (vii) x  y = (x y); (viii) (x  y)  y = (y  x)  x.

12. On one-generated projective BL-algebras LATD08Antonio Di Nola and Revaz Grigolia • Notice, that • x  y = x  y • x  y =(x  y)  y • x  y = (x y)

13. On one-generated projective BL-algebras LATD08Antonio Di Nola and Revaz Grigolia • Every MV -algebra has an underlying ordered structure x  y iff x  y = 1.

14. On one-generated projective BL-algebras LATD08Antonio Di Nola and Revaz Grigolia The following property holds in any MV -algebra: x y  x  y  x  y  x  y.

15. On one-generated projective BL-algebras LATD08Antonio Di Nola and Revaz Grigolia • The unit interval of real numbers [0,1] endowed with the following operations: x  y = min(1, x + y), x  y = max(0, x + y 1), x = 1  x, becomes an MV -algebra.

16. On one-generated projective BL-algebras LATD08Antonio Di Nola and Revaz Grigolia • It is well known that the MV -algebra S = ([0; 1];,,, 0,1) generate the variety MV of all MV -algebras, i. e. V(S) = MV. Let Q denote the set of rational numbers, for (0 )n we set Sn= (Sn; ,,, 0,1), where Sn= {0,1/n, … , n – 1/n}.

17. On one-generated projective BL-algebras LATD08Antonio Di Nola and Revaz Grigolia Let K be a variety. A algebra AK is said to be a freealgebrain K, if there exists a setA0 such that A0 generates A and every mapping f from A0 to any algebra BK is extended to a homomorphismh from A to B. In this case A0 is said to be the set of free generatorsofA. If the set of free gen

18. On one-generated projective BL-algebras LATD08Antonio Di Nola and Revaz Grigolia Also recall that an algebra AK is called projective, if for any B,CK, any epimorphism (that is an onto homomorphism )  : B C andany homomorphism  : A C, there exists a homomorphism  : A Bsuch that  = .

19. On one-generated projective BL-algebras LATD08Antonio Di Nola and Revaz Grigolia AB C   

20. On one-generated projective BL-algebras LATD08Antonio Di Nola and Revaz Grigolia McNaughton has proved that a function f : [0,1]m[0,1] has an MV polynomial representation q(x1 , . . ., xm) such that f = q iff f satisfies the following conditions:

21. On one-generated projective BL-algebras LATD08Antonio Di Nola and Revaz Grigolia • (i) f is continuous, • (ii) there exists a finite number of affine linear distinct polynomials1, . . ., s, each having the form j=bj+nj1x1+ … +njm where all b’s and n’s are integers such that for every(x1 , . . ., xm)[0, 1]mthere is j, 1 ≤ j ≤ s such that f(x1,…,xm)= j(x1,…,xm).

22. 1 On one-generated projective BL-algebras LATD08Antonio Di Nola and Revaz Grigolia 1 Green line g (x ) = x ; brown line f (x ) = 1 – x.

23. On one-generated projective BL-algebras LATD08Antonio Di Nola and Revaz Grigolia 1 1 1 1 Min(g (x), f (x )) Max(g (x), f (x ))

24. On one-generated projective BL-algebras LATD08Antonio Di Nola and Revaz Grigolia • We recall that to any 1-variable McNaughton function fis associated apartition of the unit interval [0, 1] {0 = a0,a1, … , an = 1} in such a way that a0 < a1 < … < anand the points {(a0,f(a0)), (a1,f(a1)),… , (an,f(an))} are the knots of fand the function fis linear over each interval [ai-1,ai], with i = 1, … ,n. We assume that all considered functions are 1-variableMcNaughton functions. Notice that the MV -algebra of all 1-variable Mc-Naughton functions, as a set, is closed under functional composition.

25. On one-generated projective BL-algebras LATD08Antonio Di Nola and Revaz Grigolia f (a3) f (a1) f (a2) f (a4) = f (a0) a0 a1 a2 a3 a4

26. On one-generated projective BL-algebras LATD08Antonio Di Nola and Revaz Grigolia Theorem 2. Let A be a one-generated subalgebra of FMV(1) generated by f. Then the following are equivalent: (1) A is projective; (2) one of the following holds: (2.1) Max{f(x): x  [0,1]} = f(a1) and for f non-zero function, f(x) = x for everyx  [0,a1]. (2.2) Min{f(x): x  [0,1]} = f(an1) and for f non-unit function, f(x) = x for everyx  [an1,an].

27. On one-generated projective BL-algebras LATD08Antonio Di Nola and Revaz Grigolia

28. On one-generated projective BL-algebras LATD08Antonio Di Nola and Revaz Grigolia • On Z+ we define the function v1(x) as follows: • v1(1) = 2, • v1(2) =3 2 ,… , • v1(n) = (n+1) (v1(n1) +… + v1(nk-1)), where n1(= 1), … , nk-1are all the divisors of n distinct from n(= nk). Then,

29. On one-generated projective BL-algebras LATD08Antonio Di Nola and Revaz Grigolia Let Vn denotes the variety of BL-algebras generated by (n +1)-element BL-chains. Proposition 3.(A. Di Nola, R. Grigolia, Free BL-Algebras, Proceedings of the Institute of Cybernetics, ISSN 1512-1372, Georgian Academy of Sciences, Vol. 3,N 1-2(2004), pp. 22-31). A free cyclicBL-algebra FVn(1) S1S2v1(1) … Snv1(n)  (S1 ● (S1S2v1(1) … Snv1(n)))

30. On one-generated projective BL-algebras LATD08Antonio Di Nola and Revaz Grigolia Represent FVn(1) as a direct product AnAn+, where An = S1S2v1(1) … Snv1(n) and An+= S1 ● (S1S2v1(1) … Snv1(n)). Let g(n)and g(n)+ be generatorsof An and An+, respectively. The families {An}n{0} and {An+}n{0} formdirected set of algebras with homomorphisms hij: Aj Ai and hij+: Ai+Aj+ respectively. Let A be a inverse limit of the inverse system {An}n{0} and A+ a inverse limit of the inverse system {An+}n{0}.

31. On one-generated projective BL-algebras LATD08Antonio Di Nola and Revaz Grigolia • A subalgebra A of FK(m) is said to be projective if there exists an endomorphism h : FK(m) FK(m) such that h(FK(m)) = A and h(x) = x forevery x A.

32. On one-generated projective BL-algebras LATD08Antonio Di Nola and Revaz Grigolia Proposition 4.(A. Di Nola, R. Grigolia, Free BL-Algebras, Proceedings of the Institute of Cybernetics, ISSN 1512-1372, Georgian Academy of Sciences, Vol. 3,N 1-2(2004), pp. 22-31). The subalgebraFBL(1) ofAA+ generatedby (g,g+) =((g(1),g(2), …), (g(1)+; g(2)+, …)) is one-generated free BL-algebra.

33. On one-generated projective BL-algebras LATD08Antonio Di Nola and Revaz Grigolia Theorem 4.A proper subalgebra B of one-generated free BL-algebraFBL(1) generated by(a,b)is projectiveiffb = 1 or b= g+ and the subalgebra generated by (a,1) is a projective MV -algebra.

34. On one-generated projective BL-algebras LATD08Antonio Di Nola and Revaz Grigolia PROJECTIVE FORMULAS Let us denote by Pm a set of fixed p1, … ,pm propositional variables and bym all of Basic logic formulas with variables in Pm . Notice that them-generated free BL -algebra FBL(m) is isomorphic to m/ , where  iff | () and () =() (  )). Subsequently we do notdistinguish between the formulas and their equivalence classes. Hence wesimply write m for FBL(m), and Pm plays the role of free generators. Sincem is a lattice, we have an order on m . It follows from the denition of that for all  ,  m,   iff | .

35. On one-generated projective BL-algebras LATD08Antonio Di Nola and Revaz Grigolia Let  be a formula of Basic logic and consider asubstitution  :Pm m and extend it to all of m by ((p1, … , pm)) = ((p1), … , (pm)). We can consider the substitution as an endomorphism of the free algebram. Definition 5. A formula mis called projective if there exists asubstitution  : Pm m such that |() and | (),for allm.

36. On one-generated projective BL-algebras LATD08Antonio Di Nola and Revaz Grigolia Definition 6. An algebra A is called finitely presented if A is finitely generated, withthe generatorsa1, … , amA, and there exist a finite number of equations P1(x1, … , xm) = Q1(x1, … , xm) , … ,Pn(x1, … , xm) = Qn(x1, … , xm) holdingin A on the generatorsa1, … , amAsuch that if there exists an m-generated algebra B, with generators b1, … , bmB, such that the equations P1(x1, … , xm) = Q1(x1, … , xm) , … ,Pn(x1, … , xm) = Qn(x1, … , xm) holdin B on the generatorsb1, … , bmB, then there exists a homomorphismh : AB sending ai to bi.

37. On one-generated projective BL-algebras LATD08Antonio Di Nola and Revaz Grigolia Observe that we can rewrite any equation P(x1, … , xm) = Q(x1, … , xm) in the variety BL into an equivalent one P(x1, … , xm) Q(x1, … , xm) = 1. So, for BL we can replace n equations by one /\ni =1 Pi(x1, … , xm) Qi(x1, … , xm) = 1

38. On one-generated projective BL-algebras LATD08Antonio Di Nola and Revaz Grigolia Theorem 7. A BL-algebra B is finitely presentediffBm/[u), where[u) is a principal filter generated by some elementum.

39. On one-generated projective BL-algebras LATD08Antonio Di Nola and Revaz Grigolia Theorem 8.Let A be an m-generated projective BL-algebra. Then there exists a projective formula  of m variables, such that A is isomorphic to m/[), where [) is the principal filter generated by m.

40. On one-generated projective BL-algebras LATD08Antonio Di Nola and Revaz Grigolia Corollary 9.If A is a projective MV -algebra, then A is finitely presented. Theorem 10.If  is a projective formula of m variables, then m/[)is aprojective algebra.

41. On one-generated projective BL-algebras LATD08Antonio Di Nola and Revaz Grigolia Theorem11.There exists a one-to-one correspondence between projective formulas with m variables and m-genera-ted projective subalgebras of m.