Sándor Jenei

A Proof of Standard Completeness for Esteva and Godo's Logic MTL

In the present paper we show that any at most countable linearly-ordered commutative residuated lattice can be embedded into a commutative residuated lattice on the real unit interval [0, 1]. We use this result to show that Esteva and Godos logic MTL is complete with respect to interpretations into commutative residuated lattices on [0, 1]. This solves an open problem raised in.

On the cycle-transitivity of the dice model

We introduce the notion of a dice model as a framework for describing a class of probabilistic relations. We investigate the transitivity of the probabilistic relation generated by a dice model and prove that it is a special type of cycle-transitivity that is situated between moderate stochastic transitivity or product-transitivity on the one side, and Łukasiewicz-transitivity on the other side. Finally, it is shown that any probabilistic relation with rational elements on a three-dimensional space of alternatives which possesses this particular type of cycle-transitivity, can be represented by a dice model. The same does not hold in higher dimensions.

A note on the ordinal sum theorem and its consequence for the construction of triangular norms

In this paper, the well-known ordinal sum theorem of semigroups is generalized and applied to construct new families of triangular subnorms and triangular norms (t-norms). Among them one can find several new families of left-continuous t-norms too.

Structure of left-continuous triangular norms with strong induced negations (I) Rotation construction

ABSTRACT A new algebraic construction -called rotation- is introduced in this paper which from any left-continuous triangular norm which has no zero divisors produces a left-continuous but not continuous triangular norm with strong induced negation. An infinite number of new families of such triangular norms can be constructed in this way which provides a huge spectrum of choice for e.g. logical and set theoretical connectives in non-classical logic and in fuzzy theory. On the other hand, the introduced construction brings us closer to the understanding the structure of these connectives and the corresponding logics. From the application point of view, results of this paper can be especially useful in the field of non-classical logic, fuzzy sets, and fuzzy preference modeling.

How to construct left-continuous triangular norms—state of the art☆

Left-continuity of triangular norms is the characteristic property to make it a residuated lattice. Nowadays residuated lattices are subjects of intense investigation in the fields of universal algebra and nonclassical logic. The recently known construction methods resulting in left-continuous triangular norms are surveyed in this paper.

Structure of left-continuous triangular norms with strong induced negations (II) Rotation-annihilation construction

This paper is the continuation of [11] where the rotation construction of left-continuous triangular norms was presented. Here the class of triangular subnorms and a second construction, called rotation-annihilation, are introduced: Let T1 be a left-continuous triangular norm. If T1 has no zero divisors then let T2 be a left-continuous rotation invariant t-subnorm. If T1 has zero divisors then let T2 be a left-continuous rotation invariant triangular norm. From each such pair (T1, T2) the rotation-annihilation construction produces a left-continuous triangular norm with strong induced negation. An infinite number of new families of such triangular norms can be constructed in this way, and this further extends our spectrum of choice for the proper triangular norm e.g. in probabilistic (statistical) metric spaces, or for logical and set theoretical connectives in non-classical logic, or e.g. in fuzzy sets theory and its applications. On the other hand, the introduced construction brings us closer to the understanding of the structure of these operations.

On the structure of rotation-invariant semigroups

Abstract. We generalize the notions of Girard algebras and MV-algebras by introducing rotation-invariant semigroups. Based on a geometrical characterization, we present five construction methods which result in rotation-invariant semigroups and in particular, Girard algebras and MV-algebras. We characterize divisibility of MV-algebras, and point out that integrality of Girard algebras follows from their other axioms.

Interpolation and extrapolation of fuzzy quantities – the multiple-dimensional case

Abstract This paper deals with the problem of rule interpolation and rule extrapolation for fuzzy and possibilistic systems. Such systems are used for representing and processing vague linguistic If-Then-rules, and they have been increasingly applied in the field of control engineering, pattern recognition and expert systems. The methodology of rule interpolation is required for deducing plausible conclusions from sparse (incomplete) rule bases. The interpolation/extrapolation method which was proposed for one-dimensional input space in [4] is extended in this paper to the general n-dimensional case by using the concept of aggregation operators. A characterization of the class of aggregation operators with which the extended method preserves all the nice features of the one- dimensional method is given.

On the direct decomposability of t-norms on product lattices

In this paper, a method is presented for constructing t-norms on product lattices (in other words: commutative partially ordered integral monoids over product lattices) which are not direct products. The method is fairly general and allows to generate a broad class of such t-norms. This solves an open problem posed in 1999 by De Baets and Mesiar.

A new axiomatization for involutive monoidal t-norm-based logic

On the real unit interval, the notion of a Girard monoid coincides with the notion of a t-norm-based residuated lattice with strong induced negation. A geometrical approach toward these Girard monoids, based on the notion of rotation invariance, is turned in an adequate axiomatization for the involutive monoidal t-norm-based residuated logic.