Antonio Di Nola

Fuzzy Relation Equations and Their Applications to Knowledge Engineering

1: Introductory Remarks on Fuzzy Sets.- 2: Fuzzy Relation Equations in Residuated Lattices.- 3: Lower Solutions of Max-Min Fuzzy Equations.- 4: Measures of Fuzziness of Solutions of Max-Min Fuzzy Relation Equations on Linear Lattices.- 5: Boolean Solutions of Max-Min Fuzzy Equations.- 6: ?-Fuzzy Relation Equations and Decomposable Fuzzy Relations.- 7: Max-Min Decomposition Problem of a Fuzzy Relation in Linear Lattices.- 8: Fuzzy Relation Equations with Lower and Upper Semicontinuous Triangular Norms.- 9: Fuzzy Relation Equations with Equality and Difference Composition Operators.- 10: Approximate Solutions of Fuzzy Relation Equations.- 11: Handling Fuzziness in Knowledge-Based Systems.- 12: Construction of Knowledge Base, Its Validation and Optimization.- 13: Inference Algorithms in Knowledge-Based Systems.- 14: A Fuzzy Controller and Its Realization.- 15: Bibliographies.- Author Index.

Perfect MV-algebras are categorically equivalent to abelianl-groups

In this paper we prove that the category of abelianl-groups is equivalent to the category of perfect MV-algebras. Furthermore, we give a finite equational axiomatization of the variety generated by perfect MV-algebras.

Relational equations in totally ordered lattices and their complete resolution

Abstract The set of all solutions of a composite fuzzy relation equation of Sanchez ( Inform. and Control 30 (1976)), defined on finite spaces, is studied by determining and characterizing all the lower solutions of such an equation.

Fuzzy relation equation under a class of triangular norms: A survey and new results.

By substituting the classical lattice operator min of the unit real interval with a triangular norm of Schweizer and Sklar, the usual fuzzy relational equations theory of Sanchez can be generalized to wider theory of fuzzy equations. Considering a remarkable class of triangular norms, for such type of equations defined on finite sets, we characterize the upper an lower solutions. We also characterize the solutions posessing a minimal fuzziness measure of Yager valued with respect to a triangular norm and conorm. Moreover we discuss the problem of characterization of the approximate solutions of fuzzy equations. Finally, the role of the equations considered here in creation of a formal framework for copying with fuzziness is illustrated by various examples in some well known schemes in applications of fuzzy set theory.

State-morphism MV-algebras

Abstract We present a stronger variation of state MV-algebras, recently presented by T. Flaminio and F. Montagna, which we call state-morphism MV-algebras. Such structures are MV-algebras with an internal notion, a state-morphism operator. We describe the categorical equivalences of such (state-morphism) state MV-algebras with the category of unital Abelian l -groups with a fixed state operator and present their basic properties. In addition, in contrast to state MV-algebras, we are able to describe all subdirectly irreducible state-morphism MV-algebras.

On the set of solutions of composite fuzzy relation equations

In this paper we introduce some algorithms which have minimization properties about the fuzziness of solutions in the maxmin fuzzy relation equations.

Algebraic analysis of fuzzy systems

In this paper, we have developed an algebraic theory, suitable for the analysis of fuzzy systems. We have used the notions of semiring and semimodule, introduced the notion of semilinear space, have given numerous examples of them and defined also the notions of linear dependence and independence. We have shown that the composition operation, which plays an essential role in the analysis of fuzzy systems because of its role in the compositional rule of inference, can be interpreted as a homomorphism between special semimodules. Consequently, this operation is, in a certain sense, a linear operation. This property formally explains why fuzzy systems are attractive for applications.

Compact representations of BL-algebras

Abstract.In this paper we define sheaf spaces of BL-algebras (or BL-sheaf spaces), we study completely regular and compact BL-sheaf spaces and compact representations of BL-algebras and, finally, we prove that the category of non-trivial BL-algebras is equivalent with the category of compact local BL-sheaf spaces.

On monadic MV-algebras

Abstract We define and study monadic MV -algebras as pairs of MV -algebras one of which is a special case of relatively complete subalgebra named m -relatively complete. An m -relatively complete subalgebra determines a unique monadic operator. A necessary and sufficient condition is given for a subalgebra to be m -relatively complete. A description of the free cyclic monadic MV -algebra is also given.

When is a fuzzy relation decomposable in two fuzzy sets

Two necessary and sufficient conditions are given in order to decompose an assigned fuzzy relation in two fuzzy sets.