Margarita Mas

A Survey on Fuzzy Implication Functions

One of the key operations in fuzzy logic and approximate reasoning is the fuzzy implication, which is usually performed by a binary operator, called an implication function or, simply, an implication. Many fuzzy rule based systems do their inference processes through these operators that also take charge of the propagation of uncertainty in fuzzy reasonings. Moreover, they have proved to be useful also in other fields like composition of fuzzy relations, fuzzy relational equations, fuzzy mathematical morphology, and image processing. This paper aims to present an overview on fuzzy implication functions that usually are constructed from t-norms and t-conorms but also from other kinds of aggregation operators. The four most usual ways to define these implications are recalled and their characteristic properties stated, not only in the case of [0,1] but also in the discrete case.

The distributivity condition for uninorms and t-operators

In this work we study the functional equation given by the distributivity condition: F(x, G(y,z)) = G(F(x, y),F(x,z)) for all x,y,z ∈ [0,1], where the unknown functions F,G are uninorms and/or t-operators. We characterize all the solutions in the four possible cases: (i) when F,G are t-operators, (ii) when F is a t-operator and G a uninorm, (iii) when F is a uninorm and G a t-operator and, finally, (iv) when F, G are uninorms.

t‐Operators and uninorms on a finite totally ordered set

This paper presents a detailed study of two classes of operators on a finite totally ordered set of labels L: t‐operators and uninorms. Both kinds of operators (on [0, 1]) are introduced as generalizations of t‐norms and t‐conorms. We characterize these operators on L as special combinations of operators of directed algebras in a similar way as they are characterized in the case of [0, 1] as special combinations of t‐norms and t‐conorms. We also study duality of these operators with respect to the only negation N on L, and we give the number of different t‐operators and uninorms that exist on L, related to the number of elements in L. ©1999 John Wiley & Sons, Inc.

The modularity condition for uninorms and t-operators

In this work we study the functional equation given by the modularity condition F(x,G(y,z))=G(F(x,y),z) for all z ≤ x, where the unknown functions F, G are uninorms and/or t-operators. We characterize all the solutions in the four possible cases: (i) when F, G are t-operators, (ii) when F is a t-operator and G a uninorm, (iii) when F is a uninorm and G a t-operator and, finally, (iv) when F, G are uninorms.

On the representation of fuzzy rules

In fuzzy logic, connectives have a meaning that, can frequently be known through the use of these connectives in a given context. This implies that there is not a universal-class for each type of connective, and because of that several continuous t-norms, continuous t-conorms and strong negations, are employed to represent, respectively, the and, the or, and the not. The same happens with the case of the connective If/then for which there is a multiplicity of models called T-conditionals or implications. To reinforce that there is not a universal-class for this connective, four very simple classical laws translated into fuzzy logic are studied.

Generation of weighting triangles associated with aggregation functions

In this work, we present several ways to obtain different types of weighting triangles, due to these types characterize some interesting properties of Extended Ordered Weighted Averaging operators, EOWA, and Extended Quasi-linear Weighted Mean, EQLWM, as well as of their reverse functions. We show that any quantifier determines an EOWA operator which is also an Extended Aggregation Function, EAF, i.e., the weighting triangle generated by a quantifier is always regular. Moreover, we present different results about generation of weighting triangles by means of sequences and fractal structures. Finally, we introduce a degree of orness of a weighting triangle associated with an EOWA operator. After that, we mention some results on each class of triangle, considering each one of these triangles as triangles associated with their corresponding EOWA operator, and we calculate the ornessof some interesting examples.

On left and right uninorms

This paper presents a characterization of a new family of operators. Namely, all non-decreasing, associative binary operators U: [0, 1]2 → [0, 1] with a left (or right) neutral element e ∈ [0, 1], and such that they satisfy an additional hypothesis on continuity which is called here left (or right) pseudocontinuity.

On left and right uninorms on a finite chain

The main concern of this paper is to introduce and characterize the class of operators on a finite chain L, having the same properties of pseudosmooth uninorms but without commutativity. Moreover, in this case it will only be required the existence of a one-side neutral element. These operators are characterized as combinations of AND and OR operators of directed algebras (smooth t-norms and smooth t-conorms) and the case of pseudosmooth uninorms is retrieved for the commutative case.

A characterization of (U,N), RU, QL and D-implications derived from uninorms satisfying the law of importation

The law of importation is exhaustively studied for fuzzy implications derived from uninorms. All widely used classes of implications, that is (U,N), RU, QL and D-implications derived from most usual classes of uninorms are investigated. Along this study many new solutions of the law of importation appear different from those already known for implications derived from t-norms and t-conorms.

On a class of monotonic extended OWA operators

After introducing the concepts of extended aggregation function and extended ordered weighted average (OWA) operator, we investigate the solutions of a mathematical programming problem involving this last kind of operators and from the unique solution obtained we find a class of extended OWA operators which are, in some cases, monotonic with respect to the orderings that we consider in the definition of extended aggregation function.