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Automorphisms of the algebra of fuzzy truth values II
 INT J. OF UNCERTAINTY, FUZZINESS AND KNOWLEDGEBASED SYSTEMS
, 2008
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The algebra of fuzzy truth values
 Fuzzy Sets and Systems
, 2005
"... The purpose of this paper is to give a straightforward mathematical treatment of algebras of fuzzy truth values for type2 fuzzy sets. ..."
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The purpose of this paper is to give a straightforward mathematical treatment of algebras of fuzzy truth values for type2 fuzzy sets.
Fuzzy Logics Arising from Strict De Morgan Systems
 IN PROCEEDINGS OF LINZ ’99: TOPOLOGICAL AND ALGEBRAIC STRUCTURES
, 1999
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F.: Propositional fuzzy logics: Decidable for some (algebraic) operators; undecidable for more complicated ones
 Inter. Jour. Intel. Systems
, 1999
"... If we view fuzzy logic as a logic, i.e., as a particular case of a multivalued logic, then one of the most natural questions to ask is whether the corresponding propositional logic is decidable, i.e., does there exist an algorithm that, given two propositional formulas F and G, decides whether these ..."
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Cited by 2 (1 self)
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If we view fuzzy logic as a logic, i.e., as a particular case of a multivalued logic, then one of the most natural questions to ask is whether the corresponding propositional logic is decidable, i.e., does there exist an algorithm that, given two propositional formulas F and G, decides whether these two formulas always have the same truth value. It is known that the simplest fuzzy logic, in which & = min and ∨ = max, is decidable. In this paper, we prove a more general result: that all propositional fuzzy logics with algebraic operations are decidable, We also show that this result cannot be generalized further: e.g., no deciding algorithm is possible for logics in which operations are algebraic with constructive (nonalgebraic) coefficients.
1 Varieties generated by tnorms
"... Abstract — We consider generalized tnorms on distributive lattices and investigate properties of the variety of these algebras. We compare this with the variety generated by a strict tnorm. Two algebraic systems are isomorphic if there is a onetoone mapping from one onto the other that preserves ..."
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Abstract — We consider generalized tnorms on distributive lattices and investigate properties of the variety of these algebras. We compare this with the variety generated by a strict tnorm. Two algebraic systems are isomorphic if there is a onetoone mapping from one onto the other that preserves all of the operations. If the speci ed mathematical structure of an object is all that is used in an application, then isomorphic objects are interchangeable and the choice should not in uence the quality of the model. So for many applications, the main concern is determining which systems are isomorphic and which are not. In some fuzzy logic applications, however, the key properties depend on the equations satis ed by an algebraic system. Nonisomorphic algebras can satisfy exactly the same equations, and can, for example, lead to the same propositional logic. The class of all algebras of the same type that satisfy a particular set of equations is a variety. Let I = ([0; 1] ; ^; _; 0; 1), the unit interval with minimum and maximum determined by the usual order. This is an algebra of type (2; 2; 0; 0), meaning that it has two binary operations and two nullary operations (constants). Let x 0 = 1 x. The algebras (I; 0) and
TNorms for Type2 Fuzzy Sets
"... Abstract — This paper is concerned with the de nition of tnorms on the algebra of truth values of type2 fuzzy sets. Our proposed de nition extends the de nition of ordinary tnorms on the unit interval and extends our de nition of tnorms on the algebra of truth values for intervalvalued fuzzy se ..."
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Abstract — This paper is concerned with the de nition of tnorms on the algebra of truth values of type2 fuzzy sets. Our proposed de nition extends the de nition of ordinary tnorms on the unit interval and extends our de nition of tnorms on the algebra of truth values for intervalvalued fuzzy sets. I.
Fuzzy Normal Forms and Truth Tables
"... Abstract — In this paper, we examine and compare De Morgan, Kleene, and Booleandisjunctive and conjunctive normal forms and consider their role in fuzzy settings. In particular, we show that there are normal forms and truth tables for classical fuzzy propositional logic and intervalvalued fuzzy p ..."
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Abstract — In this paper, we examine and compare De Morgan, Kleene, and Booleandisjunctive and conjunctive normal forms and consider their role in fuzzy settings. In particular, we show that there are normal forms and truth tables for classical fuzzy propositional logic and intervalvalued fuzzy propositional logic that are completely analogous to those for Boolean propositional logic. Thus, determining logical equivalence of two expressions in classical fuzzy propositional logic is a nite problem, and similarly for the intervalvalued case. Turksen's work on intervalvalued fuzzy sets is examined in light of these results. I.
Varieties generated by tnorms
"... We are concerned with the variety T of algebras of type (2,2,2,0,0) generated by the algebra (I,◦), where I = ([0,1],∧,∨,0,1) is the unit interval with minimum and maximum determined by the usual order and. ◦ = ∧ is a continuous tnorm. We have shown that a strict tnorm and a nilpotent tnorm, an ..."
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We are concerned with the variety T of algebras of type (2,2,2,0,0) generated by the algebra (I,◦), where I = ([0,1],∧,∨,0,1) is the unit interval with minimum and maximum determined by the usual order and. ◦ = ∧ is a continuous tnorm. We have shown that a strict tnorm and a nilpotent tnorm, and in fact any continuous tnorm except minimum, generate the same variety. Moreover, this variety is not generated by any finite algebra [1,2]. However, we have not determined whether or not there is a finite set of equations that determines this variety. In an attempt to answer this question, we consider the variety E of algebras of type (2,2,2,0,0) consisting of all commutative, latticeordered monoids (L,◦). By this we mean • L = (L,∧,∨,0,1) is a bounded, distributive lattice • (L,◦,1) is a commutative semigroup with identity • The semigroup operation ◦ distributes over both meet and join. The variety E is determined by a finite set of equations—namely, the equations that define a bounded, distributive lattice, together with the equations that define a commutative semigroup with identity and the equations that say ◦ distributes over both meet and join. Clearly E contains the variety generated by an algebra (I,◦) for any tnorm ◦, in particular, T ⊆ E.
1 A Family of Finite De Morgan Algebras
"... � In 1975, Zadeh proposed a setting generalizing that of both type1 and intervalvalued fuzzy sets. The truth value algebra for this new fuzzy set theory has been studied ..."
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� In 1975, Zadeh proposed a setting generalizing that of both type1 and intervalvalued fuzzy sets. The truth value algebra for this new fuzzy set theory has been studied
The Algebra of Truth Values of Type2 Fuzzy Sets: A Survey
"... Type2 fuzzy sets have come to play an increasingly important role in both applications and in the general theory of fuzzy sets. The basis of type2 fuzzy sets is a certain algebra of truth values, as set forth by Zadeh. This paper is a survey of results about this algebra, along with some new mater ..."
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Type2 fuzzy sets have come to play an increasingly important role in both applications and in the general theory of fuzzy sets. The basis of type2 fuzzy sets is a certain algebra of truth values, as set forth by Zadeh. This paper is a survey of results about this algebra, along with some new material.