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Producto, convexificación y completación de espacios métricos generalizados y probabilísticos.

Claudi Alsina — 1978


En 1967 E. Trillas introdujo la noción de espacio métrico generalizado, al considerar métricas abstractas valoradas en semigrupos ordenados, unificando con este punto de vista algebraico-reticular las estructuras métricas reales de M. Fréchet ([5]) y los espacios métricos probabilísticos de K. Menger ([6]) (así como los espacios Booleanos de Blumenthal ([4]) y las métricas naturales definidas en grupos ordenados). En el presente artículo se abordan los problemas de la topología del orden, del producto,...

Sur les mesures du degré de flou.

Enric TrillasClaudi Alsina — 1979


On caractérise toutes les entropies-floues qui sont des valuations des treillis P(X) des parties floues d'un ensemble fini X, on presente la construction de certaines entropies floues et on analyse leur caractère de valuation de treillis aiguisés Sh(g), g belonging to P(X).

On natural metrics.

Claudi AlsinaEnric Trillas — 1977


In the present note we study the effective construction of a natural generalized metric structure (on a set), obtaining as particular case the result of Menger. In the case of groups, we analyze its topology and its structure of natural proximity space (in the sense of Efremovic).

A reflection on what is a membership function.

Enric TrillasClaudi Alsina — 1999

Mathware and Soft Computing

This paper is just a first approach to the idea that the membership function μ of a fuzzy set labelled P is, basically, a measure on the set of linguistic expressions x is P for each x in the corresponding universe of discourse X. Estimating that the meaning of P (relatively to X) is nothing else than the use of P on X, these measures seem to be reached by generalizing to a preordered set the concept of Fuzzy Measure, introduced by M. Sugeno, when the preorder translates the primary use of the predicate...

Didactical note: probabilistic conditionality in a Boolean algebra.

Enric TrillasClaudi AlsinaSettimo Termini — 1996

Mathware and Soft Computing

This note deals with two logical topics and concerns Boolean Algebras from an elementary point of view. First we consider the class of operations on a Boolean Algebra that can be used for modelling If-then propositions. These operations, or Conditionals, are characterized under the hypothesis that they only obey to the Modus Ponens-Inequality, and it is shown that only six of them are boolean two-place functions. Is the Conditional Probability the Probability of a Conditional? This problem will...

Combining degrees of impairment: the case of the index of Balthazard.

Claudi AlsinaEnric TrillasClaudio Moraga — 2003

Mathware and Soft Computing

Using techniques for modeling indices by means of functional equations and resources from fuzzy set theory, the classical Balthazard index used in order to combine several degrees of impairment is characterized in two natural ways and its use is criticized. In addition some hints are given on how to study better solutions than Balthazard's one for the problem of combining impairment degrees.

On MPT-implication functions for fuzzy logic.

Enric TrillasClaudi AlsinaAna Pradera — 2004


This paper deals with numerical functions J : [0,1] x [0,1] → [0,1] able to functionally express operators →: [0,1] x [0,1] → [0,1] defined as (μ → σ)(x,y) = J(μ(x),σ(y)), and verifying either Modus Ponens or Modus Tollens, or both. The concrete goal of the paper is to search for continuous t-norms T and strong-negation functions N for which it is either T(a, J(a,b)) ≤ b (Modus Ponens) or T(N(b), J(a,b)) ≤ N(a) (Modus Tollens), or both, for all a,b in [0,1] and a given J. Functions J are taken among...

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