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The study of consistency in decision-making models is of great importance. Our proposal deals with a special type of extended aggregation functions to guarantee consistency that is based on T-S consistency of linearly ordered fuzzy preference relations (fuzzy ranking for preferences). Necessary conditions to ensure consistency in the aggregation process are provided. Moreover, sufficient conditions to guarantee the existence of extended aggregation functions verifying T-S consistency are presented. Some examples of functions of this type are also given. The equivalence between T-S consistency and the additive consistency for fuzzy preference relations is established.

Starting from the notion of a multidistance, we formalize, through a suitable system of axioms, the concept of an inequality measure defined on a nonempty set with no additional structure implemented a priori. Among inequality measures, apart from multidistances we pay special attention to dispersions, and study their main features. Classical concepts will be generalized to this abstract setting. Multidistances are then revisited, and some new methods to generate them are implemented. A wide spectrum of interdisciplinary applications is outlined in the final section. This work is partially supported by the research projects ECO2015-65031-R , MTM2015-63608-P , TIN2016-77356-P , TIN2017-87600-P (MINECO/AEI-FEDER, UE) and PID2019-108392GB-I00 (AEI/10.13039/501100011033)

In 1981, J. Borsik and J. Dobos analyzed what properties a function must satisfy in order to merge a collection of metric spaces into a single one. Later on, E. Castineira, A. Pradera and E. Trillas studied a variant of the same problem in which each metric of the collection to be merged is defined on the same non-empty set. In this, paper we continue the work in this last aforesaid direction. On the one hand, we provide a new characterization of such functions and a few methods to construct them. On the other hand, we analyze the existence of absorbent, idempotent and neutral elements for such class of functions and, thus, we design techniques that allow to discard those functions that are not useful for merging metrics. Finally, we discuss when the functions under study preserve metrics.

The problem of the consistency of qualitative judgements about the difference of attractiveness between alternatives is studied. Our proposal deals with smooth t-conorms to be used for aggregation of elementary judgements.

The construction of distances from t-norms is studied. The generation of distances from a t-norm and its dual t-conorm is extended to the more general case of any t-norm and t-conorm. The open problem of characterizing those t-norms that define distances is solved in the case that the t-norms have the same zero region as the Lukasiewicz one.

In the paper we introduce and study quadratic constructions of copulas based on composition of a copula and some quadratic polynomial. We characterize all quadratic polynomials whose composition with an arbitrary copula always results in a copula. Due to this result, we can assign a two-parametric class K C c , d of copulas with parameters ( c , d ) in a certain subset ? ? R 2 to each copula C. Moreover, we also determine all copulas invariant with respect to the quadratic constructions presented. This investigation brings two interesting parametric classes of copulas. We show that the union of these two classes is equal to the so-called Plackett family of copulas. We add some properties of these copulas and their statistical consequences.

This work is developed in the field of the additive generation of discrete aggregation operators. Specifically, this article deals with the study and applicability of disjunctions that are additively generated by concave, convex and symmetric generators.

Fuzzy operations acting on entire fuzzy sets with the stress on fuzzy implications are discussed and studied. In the case of binary operations, the input fuzzy sets are fuzzy subsets of possibly different universal spaces X and Y, and the output fuzzy set is a fuzzy subset of the Cartesian product Xi?Y. The standard approach to fuzzy operations is based on functions acting on 0,1], and then these fuzzy operations are called functionally expressible. We give a characterization of functionally expressible fuzzy implications (and other fuzzy operations), and include several examples of fuzzy operations which are not functionally expressible. An axiomatic approach for fuzzy operations, specially fuzzy implications, is presented.These fuzzy operations act on entire input fuzzy sets which can belong to possible different universal spaces.A characterization of functionally expressible fuzzy implications is given.Several examples of fuzzy operations which are not functionally expressible are presented.

The problem of how to merge, by means of a function, a family of metrics into a single one was studied deeply by J. Borsik and J. Dobos [On a product of metric spaces, Math. Slovaca31 (1981) 193-205]. Motivated by the utility of quasi-metrics in Computer Science, the Borsik and Dobos study was extended to the quasi-metric context in such a way that a general description of how to combine through a function a family of quasi-metrics in order to obtain a single one as output was provided by G. Mayor and O. Valero [Aggregation of asymmetric distances in Computer Science, Inform. Sci.180 (2010) 803-812]. In this paper, inspired by the fact that fixed point theory provides an efficient tool in many fields of applied sciences, we have proved fixed point theorems for a new type of contractions, that we have called projective @F-contractions, defined between quasi-metric spaces that have been obtained via the so-called quasi-metric aggregation functions. Moreover, we show that the new fixed point results are useful to discuss, on the one hand, the complexity of a collection of recursive programs whose running times of computing hold a coupled system of recurrence equations and, on the other hand, to analyze simultaneously the complexity and the correctness of recursive algorithms that perform a computation by means of a recursive denotational specification.

The main purpose of this paper is to solve the functional equation T + λ S = Min + λ Max , for 0 ≤ λ ≤ ∞ and a pair ( T , S ) of a t-norm T and a t-conorm S. This equation arises when we consider a convex linear combination of a t-norm and a t-conorm, and set out the problem of finding the intersection of the segments determined by the pairs ( Min , Max ) and ( T , S ) .

In this contribution we deal with a global monotonicity condition for the class of associative extended aggregation functions. We insist on the idea that global monotonicity can be taken as a minimum requirement for an extended aggregation function to be considered consistent.

After introducing a definition of dispersion measure on the Euclidean space \(\mathbb {R}^k\), we deal with the connection between these measures and the so called multidistances. In this way, we show that thr standard deviation is a relevant example of multidistance and, on the other hand, several significant families of multidistances are, at the same time, dispersion measures. Sufficient conditions for a multidistance to be a dispersion measure are also established.

In 1981, Borsik and Dobos studied in depth the problem of how to merge, by means of a function, a collection (not necessarily finite) of distance spaces in order to obtain a single one as a result [J. Borsik, J. Dobos, On a product of metric spaces, Math. Slovaca 31 (1981) 193-205]. Later on, Herburt and Moszynska studied the same problem for the case of normed linear spaces, inspired by the fact that every norm induces in a natural way a distance on a linear space, and analyzed the relationship between the both aforenamed problems [I. Herburt, M. Moszynska, On metric products, Colloq. Math. 62 (1991) 121-133]. More recently, Romaguera and Schellekens introduced a mathematical approach, based on the notions of asymmetric distance and asymmetric normed linear space, which is suitable for the complexity analysis of programs and algorithms in Computer Science [S. Romaguera, M. Schellekens, Quasi-metric properties of complexity spaces, Topology Appl. 98 (1999) 311-322]. In this paper, motivated by the importance of the information fusion techniques in Artificial Intelligence and by the utility of asymmetric distances and asymmetric norms in Computer Science, we study the Herburt and Moszynska problem for asymmetric normed linear spaces. In particular we give a general description of how to combine a collection (not necessarily finite) of asymmetric normed linear spaces in order to obtain a single one as output and, in addition, we clear up the relationship between this problem and its analogous of combining asymmetric distance spaces which has been already explored by Mayor and Valero [G. Mayor, O. Valero, Aggregation of asymmetric distances in computer science, Inform. Sci. 180 (2010) 803-812]. Furthermore, it is shown that the asymmetric norms employed, in the spirit of Romaguera and Schellekens, in complexity analysis can be retrieved as a particular case of the developed theory. The last fact opens the possibility of applying a wide range of properties from the general aggregation theory in Artificial Intelligence to the complexity analysis of programs and algorithms in Computer Science.

In this paper, we propose a formal definition of a multi-argument function distance. The conventional definition of a distance is extended to apply to collections of more than two elements. Some general properties of multidistances are investigated and significant examples are exhibited. Finally, we introduce the concept of ball centered at a list in order to extend the one of the usual ball in a metric space.

In this paper we provide a general description of how to combine a collection (not necessarily finite) of asymmetric distances in order to obtain a single one as output. To this end we introduce the notion of asymmetric distance aggregation function that generalizes the well-known one for distance spaces given by Borsik and Dobos [J. Borsik, J. Dobos, On a product of metricspaces, Math. Slovaca 31 (1981) 193-205]. Among other results, a characterization of such functions is obtained in terms of monotony and subadditivity. Finally, we relate our results to Computer Science. In particular we show that the mathematical formalism based on complexity distances, which has been introduced by Romaguera and Schellekens [S. Romaguera, M. Schellekens, Quasi-metric properties of complexity spaces, Topol. Appl. 98 (1999) 311-322] for the complexity analysis of programs and algorithms, can be obtained as a particular case of our new framework using appropriate asymmetric aggregation distance functions.

This paper deals with triangular norms and conorms defined on the extended set N of natural numbers ordered by divisibility. From the fundamental theorem of arithmetic, N can be identified with a lattice of functions from the set of primes to the complete chain {0, 1, 2, …, +∞}, thus our knowledge about (divisible) t-norms on this chain can be applied to the study of t-norms on N. A characterization of those t-norms on N which are a direct product of t-norms on {0, 1, 2, …, +∞} is given and, after introducing the concept of T-prime (prime with respect to a t-norm T), a theorem about the existence of a T-prime decomposition is obtained. This result generalizes the fundamental theorem of arithmetic.

Multisets are set-like structures where an element can appear more than once. They are also called bags. A set means a collection of types of objects {x,y,...} rather than of concrete tokens{x,x,x,y,y,...} of them. The set of multisets on a universe is a partially ordered set for a functionally defined relationship of order. Moreover, it is a product of chains.Several pointwise defined operations as the addition, the union and the intersection between multisets have been defined and their properties investigated in several papers. The union and the intersection of two multisets is defined by means of the maximum and the minimum of the respective functions in N=N∪{∞} and a lattice structure is obtained for the previously defined poset.But, for multisets, the addition is an important operation because it corresponds to the simultaneous consideration of two multisets on a universe (or the consideration on a multiset twice). The addition can be defined as the disjoint union, i.e. A+B=A⊎B and of course is not idempotent. The addition and the union satisfy the properties of t-conorms in the same way that the intersection is a t-norm on the poset of the multisets and they are functionally, even pointwise, defined. In this paper we are concerned with some representation of all the possible functionally defined t-norms and t-conorms over the poset of the multisets that satisfy some interesting property like the divisibility. We distinguish the cases where the multisets are bounded or not.

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.

This paper deals with the well-known Sklar's theorem, which shows how joint distribution functions are related to their marginals by means of copulas. The main goal is to prove a discrete version of this theorem involving copula-like operators defined on a finite chain, that will be called discrete copulas. First, the idea of subcopulas in this finite setting is introduced and the problem of extending a subcopula to a copula is solved. This is precisely the key point which allows to state and prove the discrete version of Sklar's theorem.

The operators defined on a finite chain, usually called discrete operators, constitute a field of increasing interest because of their applications to qualitative aggregation and computing with words. We introduce in this paper the residual implications derived from discrete uninorms, we look at their most interesting properties and their axiomatic characterization is also given.

Once introduced a definition of symmetric difference function on the unit real interval [0,1], we consider a method to construct such functions based on a triplet formed by a t-norm, a t-conorm and a strong negation. Our main goal is to characterize those triplets that define symmetric difference functions which are distances.

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.

In this paper, a new approach to the study of binary operations is shown. After introducing the concept of external range, some results concerning the class of continuous binary operations with finite external range are given.