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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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On one-generated projective BL-algebras LATD08 Antonio Di Nola and Revaz Grigolia
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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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
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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.
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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 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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On one-generated projective BL-algebras LATD08 Antonio Di Nola and Revaz Grigolia
According to the definition we have
B1 B2
B1● B2
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On one-generated projective BL-algebras LATD08 Antonio Di Nola and Revaz Grigolia
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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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.
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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,
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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.
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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)
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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.
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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.
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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.
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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
Sn = (Sn; ,, , 0,1),
where Sn = {0,1/n, … , n – 1/n}.
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On one-generated projective BL-algebras LATD08 Antonio Di Nola and Revaz Grigolia
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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On one-generated projective BL-algebras LATD08 Antonio Di Nola and Revaz Grigolia
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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On one-generated projective BL-algebras LATD08 Antonio Di Nola and Revaz Grigolia
A B
C
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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(x1 , . . ., xm)
such that f = q iff f
satisfies the following conditions:
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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=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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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.
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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
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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 = 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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On one-generated projective BL-algebras LATD08 Antonio Di Nola and Revaz Grigolia
a0 a1
a2 a3 a4
f (a4) = f (a0)
f (a1)
f (a2)
f (a3)
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On one-generated projective BL-algebras LATD08 Antonio 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 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 one-generated projective BL-algebras LATD08 Antonio Di Nola and Revaz Grigolia
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On one-generated projective BL-algebras LATD08 Antonio 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-1 are all the divisors of n distinct from n(= nk).
Then,
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On one-generated projective BL-algebras LATD08 Antonio 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 cyclic BL-algebra
FVn(1) S1 S2 v1(1) … Snv1(n)
(S1 ● (S1 S2 v1(1) … Snv1(n) ))
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On one-generated projective BL-algebras LATD08 Antonio Di Nola and Revaz Grigolia
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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On one-generated projective BL-algebras LATD08 Antonio 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 for every x A.
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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 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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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 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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On one-generated projective BL-algebras LATD08 Antonio 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 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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On one-generated projective BL-algebras LATD08 Antonio Di Nola and Revaz Grigolia
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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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 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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On one-generated projective BL-algebras LATD08 Antonio 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
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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 .
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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 .
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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.
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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 .