1 On one-generated projective BL-algebras Antonio Di Nola and Revaz Grigolia University of Salerno...

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On one-generated projective BL-algebras

Antonio Di Nola and Revaz Grigolia

University of Salerno Tbilisi State UniversityDepartment of Mathematics Institute of Cybernetics

and Informatics .

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

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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.

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BasicBasic 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 *.

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• 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

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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.

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• 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 isomorphic copies of the ones assume that 1B1 = 0B2 and

(B1 \ {1B1 }) (B2 \ {0B2 }) = .

• Let B1● B2 be the structure whose universe is B1 B2 and x y if

(x, y B1 and x 1 y) or (x, y B2 and x 2 y), or x B1 and y B2 . Moreover,

• x y = x i y for x, y Bi, x y = x for x B1 and y B2 ;

• x y = 1B2 for x y;

• for x > y we put x y = x i y if x, y Bi and

• put x y = y for x B2 and y B1 \ B2.

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According to the definition we have

B1 B2

B1● B2

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Proposition 1. B = B1● B2 is a BL-algebra with 0B = 0B1 ; 1B = 1B2 and 1B1 = 0B2 being non-extremal idempotent. Moreover, if B1,B2 are BL-chains, then B = B1● B2 is BL-chain too.

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• 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.

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• A BL-algebra is said to be an MV -algebra, if it satises the following equation:

x = x,

where x = x 0.

More precisely,

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• 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.

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• Notice, that

• x y = x y• x y = (x y) y • x y = (x y)

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• Every MV -algebra has an underlying ordered structure

x y iff x y = 1.

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The following property holds in any

MV -algebra:

x y x y x y x y.

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• 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.

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• 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}.

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Let K be a variety. A algebra AK is said to be a free algebra in K, if there exists a set A0 such that A0 generates A and every

mapping f from A0 to any algebra BK is extended to a homomorphism h from A to B. In this case A0 is said to be the set of free generators of A. If the set of free gen

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Also recall that an algebra AK is called projective, if for any B,CK, any epimorphism (that is an onto homomorphism ) : B C and any homomorphism : A C, there exists a homomorphism : A B such that

= .

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A B

C

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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:

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• (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]m there is j, 1 ≤ j ≤ s such that

f(x1,…,xm)= j (x1,…,xm).

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1

1

Green line g (x ) = x ; brown line f (x ) = 1 – x.

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Min(g (x), f (x ))

1

1

Max(g (x), f (x ))

1

1

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• We recall that to any 1-variable McNaughton function f is associated a partition of the unit interval [0, 1]

{0 = a0, a1, … , an = 1} in such a way that

a0 < a1 < … < an and the points

{(a0, f(a0)), (a1, f(a1)), … , (an, f(an))}

are the knots of f and the function f is linear over each interval

[ai -1, ai ],

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.

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a0 a1

a2 a3 a4

f (a4) = f (a0)

f (a1)

f (a2)

f (a3)

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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 every x [0,a1].

(2.2) Min{f(x): x [0,1]} = f(an 1) and for f non-unit function, f(x) = x for every x [an 1, an].

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• 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-1 are all the divisors of n distinct from n(= nk).

Then,

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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 cyclic BL-algebra

FVn(1) S1 S2 v1(1) … Snv1(n)

(S1 ● (S1 S2 v1(1) … Snv1(n) ))

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Represent FVn(1) as a direct product An An+ , where

An = S1 S2 v1(1) … Snv1(n) and

An+ = S1 ● (S1 S2 v1(1) … Sn

v1(n) ).

Let g(n) and g(n)+ be generators of An and An+, respectively.

The families {An}n{0} and {An+}n{0}

form directed 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}.

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• 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 for every x A.

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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 subalgebra FBL(1) of A A+ generated by

(g, g+) = ((g(1), g(2), …), (g(1)+; g(2)+, …))

is one-generated free BL-algebra.

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Theorem 4. A proper subalgebra B of one-generated free BL-algebra FBL(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.

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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 the m-generated free BL -algebra FBL(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 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 | .

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Let be a formula of Basic logic and consider a substitution

: 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 m is called projective if there exists a substitution

: Pm m

such that

| ( ) and | () , for all m .

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Definition 6. An algebra A is called finitely presented if A

is finitely generated, with the generators a1, … , am A, and there

exist a finite number of equations

P1(x1, … , xm) = Q1(x1, … , xm) , … , Pn(x1, … , xm) = Qn(x1, … , xm)

holding in A on the generators a1, … , am A such 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)

hold in B on the generators b1, … , bm B, then there exists a

homomorphism h : A B sending ai to bi.

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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

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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 .

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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 .

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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.

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Theorem 11. There exists a one-to-one correspondence between projective

formulas with m variables and m-genera-ted projective subalgebras of m .

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