1 On one-generated projective BL-algebras Antonio Di Nola and Revaz Grigolia University of Salerno Tbilisi State University Department of Mathematics Institute of Cybernetics and Informatics. Logic, Algebra and Truth Degrees 2008 September 8 to 11, Siena, Italy

2 On one-generated projective BL-algebras LATD08 Antonio 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 LATD08 Antonio Di Nola and Revaz Grigolia Basic 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 LATD08 Antonio 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 LATD08 Antonio 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 commutative monoid 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 LATD08 Antonio 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 B 1, B 2 be BL-algebras, where B 1 is BL-chain. Taking isomorphic copies of the ones assume that 1 B 1 = 0 B 2 and (B 1 \ {1 B 1 })  (B 2 \ {0 B 2 }) = . Let B 1 ● B 2 be the structure whose universe is B 1  B 2 and x  y if (x, y  B 1 and x  1 y) or (x, y  B 2 and x  2 y), or x  B 1 and y  B 2. Moreover, x  y = x  i y for x, y  B i, x  y = x for x  B 1 and y  B 2 ; x  y = 1 B 2 for x  y; for x > y we put x  y = x  i y if x, y  B i and put x  y = y for x  B 2 and y  B 1 \ B 2.

7 On one-generated projective BL-algebras LATD08 Antonio Di Nola and Revaz Grigolia According to the definition we have B1B1 B2B2 B1● B2B1● B2

8 On one-generated projective BL-algebras LATD08 Antonio Di Nola and Revaz Grigolia Proposition 1. B = B 1 ● B 2 is a BL-algebra with 0 B = 0 B 1 ; 1 B = 1 B 2 and 1 B 1 = 0 B 2 being non-extremal idempotent. Moreover, if B 1,B 2 are BL-chains, then B = B 1 ● B 2 is BL-chain too.

9 On one-generated projective BL-algebras LATD08 Antonio Di Nola and Revaz Grigolia The variety BL of all BL-algebras is not locally finite and it is generated by 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 LATD08 Antonio Di Nola and Revaz Grigolia A BL-algebra is said to be an MV -algebra, if it satises the following equation:  x = x, where  x = x  0. More precisely,

11 On one-generated projective BL-algebras LATD08 Antonio Di Nola and Revaz Grigolia An algebra A = (A; , , , 0,1), is said to be an MV-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 LATD08 Antonio 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 LATD08 Antonio 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 LATD08 Antonio 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 LATD08 Antonio 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 LATD08 Antonio 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 S n = (S n ; , , , 0,1), where S n = {0,1/n, …, n – 1/n}.

17 On one-generated projective BL-algebras LATD08 Antonio Di Nola and Revaz Grigolia Let K be a variety. A algebra A  K is said to be a free algebra in K, if there exists a set A 0 such that A 0 generates A and every mapping f from A 0 to any algebra B  K is extended to a homomorphism h from A to B. In this case A 0 is said to be the set of free generators of A. If the set of free gen

18 On one-generated projective BL-algebras LATD08 Antonio 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 and any homomorphism  : A  C, there exists a homomorphism  : A  B such that   = .

19 On one-generated projective BL-algebras LATD08 Antonio Di Nola and Revaz Grigolia A B C   

20 On one-generated projective BL-algebras LATD08 Antonio Di Nola and Revaz Grigolia McNaughton has proved that a function f : [0,1] m  [0,1] has an MV polynomial representation q(x 1,..., x m ) such that f = q iff f satisfies the following conditions:

21 On one-generated projective BL-algebras LATD08 Antonio 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 =b j +n j 1 x 1 + … +n j m where all b’s and n’s are integers such that for every (x 1,..., x m )  [0, 1] m there is j, 1 ≤ j ≤ s such that f(x 1,…,x m )= j (x 1,…,x m ).

22 On one-generated projective BL-algebras LATD08 Antonio Di Nola and Revaz Grigolia 1 1 Green line g (x ) = x ; brown line f (x ) = 1 – x.

23 On one-generated projective BL-algebras LATD08 Antonio Di Nola and Revaz Grigolia Min(g (x), f (x )) 1 1 Max(g (x), f (x )) 1 1

24 On one-generated projective BL-algebras LATD08 Antonio Di Nola and Revaz Grigolia We recall that to any 1-variable McNaughton function f is associated a partition of the unit interval [0, 1] {0 = a 0, a 1, …, a n = 1} in such a way that a 0 < a 1 < … < a n and the points {(a 0, f(a 0 )), (a 1, f(a 1 )), …, (a n, f(a n ))} are the knots of f and the function f is linear over each interval [a i -1, a i ], with i = 1, …, n. We assume that all considered functions are 1-variable McNaughton 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 LATD08 Antonio Di Nola and Revaz Grigolia a0a0 a1a1 a2a2 a3a3 a4a4 f (a 4 ) = f (a 0 ) f (a 1 ) f (a 2 ) f (a 3 )

26 On one-generated projective BL-algebras LATD08 Antonio Di Nola and Revaz Grigolia Theorem 2. Let A be a one-generated subalgebra of F MV (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(a 1 ) and for f non- zero function, f(x) = x for every x  [0,a 1 ]. (2.2) Min{f(x): x  [0,1]} = f(a n  1 ) and for f non- unit function, f(x) = x for every x  [a n  1, a n ].

27 On one-generated projective BL-algebras LATD08 Antonio Di Nola and Revaz Grigolia

28 On one-generated projective BL-algebras LATD08 Antonio Di Nola and Revaz Grigolia On Z+ we define the function v 1 (x) as follows: v 1 (1) = 2, v 1 (2) = 3  2,…, v 1 (n) = (n+1)  (v 1 (n 1 ) +… + v 1 (n k-1 )), where n 1 (= 1), …, n k-1 are all the divisors of n distinct from n(= n k ). Then,

29 On one-generated projective BL-algebras LATD08 Antonio Di Nola and Revaz Grigolia Let V n 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 , Georgian Academy of Sciences, Vol. 3,N 1-2(2004), pp ). A free cyclic BL-algebra F Vn (1)  S 1  S 2 v 1 (1)  …  S n v 1 (n)   (S 1 ● (S 1  S 2 v 1 (1)  …  S n v 1 (n) ))

30 On one-generated projective BL-algebras LATD08 Antonio Di Nola and Revaz Grigolia Represent F Vn (1) as a direct product A n  A n +, where A n = S 1  S 2 v 1 (1)  …  S n v 1 (n) and A n + = S 1 ● (S 1  S 2 v 1 (1)  …  S n v 1 (n) ). Let g (n) and g (n)+ be generators of A n and A n +, respectively. The families {A n } n  {0} and {A n + } n  {0} form directed set of algebras with homomorphisms h ij : A j  A i and h ij + : A i +  A j + respectively. Let A be a inverse limit of the inverse system {A n } n  {0} and A + a inverse limit of the inverse system {A n + } n  {0}.

31 On one-generated projective BL-algebras LATD08 Antonio Di Nola and Revaz Grigolia A subalgebra A of F K (m) is said to be projective if there exists an endomorphism h : F K (m)  F K (m) such that h(F K (m)) = A and h(x) = x for every x  A.

32 On one-generated projective BL-algebras LATD08 Antonio Di Nola and Revaz Grigolia Proposition 4. (A. Di Nola, R. Grigolia, Free BL-Algebras, Proceedings of the Institute of Cybernetics, ISSN , Georgian Academy of Sciences, Vol. 3,N 1-2(2004), pp ). The subalgebra F BL (1) of A  A + generated by (g, g + ) = ((g (1), g (2), …), (g (1)+ ; g (2)+, …)) is one-generated free BL-algebra.

33 On one-generated projective BL-algebras LATD08 Antonio Di Nola and Revaz Grigolia Theorem 4. A proper subalgebra B of one-generated free BL-algebra F BL (1) generated by (a,b) is projective iff b = 1 or b = g + and the subalgebra generated by (a,1) is a projective MV -algebra.

34 On one-generated projective BL-algebras LATD08 Antonio Di Nola and Revaz Grigolia PROJECTIVE FORMULAS Let us denote by P m a set of fixed p 1, …, p m propositional variables and by  m all of Basic logic formulas with variables in P m. Notice that the m-generated free BL -algebra F BL (m) is isomorphic to  m / , where    iff |  (    ) and (    ) =(    )  (    )). Subsequently we do not distinguish between the formulas and their equivalence classes. Hence we simply write  m for F BL (m), and P m 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 LATD08 Antonio Di Nola and Revaz Grigolia Let  be a formula of Basic logic and consider a substitution  : P m   m and extend it to all of  m by  (  (p 1, …, p m )) =  (  (p 1 ), …,  (p m )). We can consider the substitution as an endomorphism of the free algebra  m. Definition 5. A formula    m is called projective if there exists a substitution  : P m   m such that |   (  ) and  |     (  ), for all    m.

36 On one-generated projective BL-algebras LATD08 Antonio Di Nola and Revaz Grigolia Definition 6. An algebra A is called finitely presented if A is finitely generated, with the generators a 1, …, a m  A, and there exist a finite number of equations P 1 (x 1, …, x m ) = Q 1 (x 1, …, x m ), …, P n (x 1, …, x m ) = Q n (x 1, …, x m ) holding in A on the generators a 1, …, a m  A such that if there exists an m-generated algebra B, with generators b 1, …, b m  B, such that the equations P 1 (x 1, …, x m ) = Q 1 (x 1, …, x m ), …, P n (x 1, …, x m ) = Q n (x 1, …, x m ) hold in B on the generators b 1, …, b m  B, then there exists a homomorphism h : A  B sending a i to b i.

37 On one-generated projective BL-algebras LATD08 Antonio Di Nola and Revaz Grigolia Observe that we can rewrite any equation P(x 1, …, x m ) = Q(x 1, …, x m ) in the variety BL into an equivalent one P(x 1, …, x m )  Q(x 1, …, x m ) = 1. So, for BL we can replace n equations by one /\ n i =1 P i (x 1, …, x m )  Q i (x 1, …, x m ) = 1

38 On one-generated projective BL-algebras LATD08 Antonio Di Nola and Revaz Grigolia Theorem 7. A BL-algebra B is finitely presented iff B   m /[u), where [u) is a principal filter generated by some element u   m.

39 On one-generated projective BL-algebras LATD08 Antonio 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 LATD08 Antonio 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 a projective algebra.

41 On one-generated projective BL-algebras LATD08 Antonio Di Nola and Revaz Grigolia Theorem 11. There exists a one-to-one correspondence between projective formulas with m variables and m-genera- ted projective subalgebras of  m.