Gaspar Mayor

On a class of operators for expert systems

We use the concept of directed algebra (closely related to De Morgan triplets) to modelize connectives in expert systems when linguistic terms are introduced. Mainly this article describes all directed algebra structures on a totally ordered finite set.

On locally internal monotonic operations

This paper deals with monotonic binary operations F: [0, 1]2 → [0, 1] with the property (called locally internal property) that the value at any point (x, y) is always one of its arguments x, y. After stating a theorem that characterizes this kind of operations, some special cases are studied in detail by considering additional properties of the operation: commutativity, existence of a neutral element and associativity. In case of locally internal, associative monotonic operations with neutral element, a characterization theorem gives an improvement of a well-known theorem of Czogala and Drewniak on idempotent, associative and increasing operations with neutral element, as well as an improvement of a characterization theorem for left (and right) continuous, idempotent uninorms.

7 – Triangular norms on discrete settings

Triangular norms and conorms, as well as copulas acting on finite or infinite discrete chains are studied. Special attention is paid to the divisibility of these operations.

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.

Weighted ordinal means

The concept of weighted ordinal arithmetic means and other related weighted ordinal means is studied. Based on the relevant results on the scale [0,1], new types of weighted ordinal means are proposed. In some cases these ordinal means coincide with those proposed by Godo and Torra, but not in the case when ordinal means introduced by them are not idempotent. Based on divisible ordinal t-conorms and modifying the approach of Godo and Torra, we show how the previously introduced weighted ordinal means can be obtained without exploiting the formal similarity of the structure of continuous t-conorms on [0,1] and divisible ordinal t-conorms.

Copula-like operations on finite settings

This paper deals with discrete copulas considered as a class of binary aggregation operators on a finite chain. A representation theorem by means of permutation matrices is given. From this characterization, we study the structure of associative discrete copulas and a theorem of decomposition of any discrete copula in terms of associative discrete copulas is obtained. Finally, some aspects concerning their extension to copulas are dealt with.

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 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.