Catálogo de publicaciones - libros

We analyze differences between –algebras and –algebras. The study has application in mathematical fuzzy logic as the Lindenbaum algebras of Lukasiewicz logic or Hájek’s –logics are –algebras or –algebras, respectively. We focus on possible generalizations of Boolean elements of a general –algebra ; we prove that an element  ∈  is Boolean iff  ∨  = . is called semi–Boolean if, for all  ∈ , is Boolean. We prove that an –algebra is semi–Boolean iff is a Boolean algebra. A –algebra is semi–Boolean iff is a –algebra. A –algebra L is called hyper–Archimedean if, for all  ∈ , there is an  ≥ 1 such that is Boolean. We prove that hyper–Archimedean –algebras are –algebras. We discuss briefly the applications of our results in mathematical fuzzy logic.

A fuzzy version of the Hahn-Banach theorem is proved based on the classical result. A comparison is also drawn with an earlier published result in this connection.

From logic and algebra point of view, Computing with words is discussed in this paper. By analyzing the semantically ordering relation of linguistic variable , orderings on linguistic hedges and atomic evaluating syntagm  = {, } are obtained, respectively. Let be finite chain, then Lukasiewicz product algebra of  = × of is obtained, and term-set () of is embedded into an algebra of type  = { ∨ , ∧ , ′,→}. In some cases, can be applied in linguistic decision directly, also as truth domain of logic statements. Different with other truth domain, here truth values are linguistic terms rather than numerals (or symbolic).

Let be any subgroup of the group of Möbius transformations and a set of stabilizers and their intersections. Taking a fuzzy subgroup of given by means of stabilizers of as the mapping , we examine the behaviour of the meet operation ∧ in .

Estimation of membership function is one of the most important problems in the application of fuzzy sets. This paper presents one of approaches to this problem. A method for estimation of membership function is proposed, based on fuzzy measures: fuzzy entropy and fuzzy index. Examples of generating membership function in the field of image processing are shown.The method presented in this paper can be used in other fields of computer sciences, where statistical data are available.

We study finite residuated lattices with up to 11 elements. We present an algorithm for generating all non-isomorphic finite residuated lattices with a given number of elements. Furthermore, we analyze selected properties of all the lattices generated by our algorithm and present summarizing statistics.

The paper develops fuzzy attribute logic, i.e. a logic for reasoning about formulas of the form where and are fuzzy sets of attributes. A formula represents a dependency which is true in a data table with fuzzy attributes iff each object having all attributes from has also all attributes from , membership degrees in and playing a role of thresholds. We study axiomatic systems of fuzzy attribute logic which result by adding a single deduction rule, called a rule of multiplication, to an ordinary system of deduction rules complete w.r.t. bivalent semantics, i.e. to well-known Armstrong axioms. In this paper, we concentrate on the rule of multiplication and its role in fuzzy attribute logic. We show some advantageous properties of the rule of multiplication. In addition, we show that these properties enable us to reduce selected problems concerning proofs in fuzzy attribute logic to the corresponding problems in the ordinary case. As an example, we discuss the problem of normalization of proofs and present, in the setting of fuzzy attribute logic, a counterpart to a well-known theorem from database theory saying that each proof can be transformed to a so-called RAP-sequence.

In this contribution, we will recall graded fuzzy rules introduced in [5] and explain the difference from the classical fuzzy rules. Moreover, properties of formulae, which are used to formalize the graded fuzzy rules, will be recalled.

The procedure of evaluating the results of a clustering algorithm is know under the term cluster validity. In general terms, cluster validity criteria can be classified in three categories: internal, external and relative. In this work we focus on the external criteria, which evaluate the results of a clustering algorithm based on a pre-specified structure , that pertains to the data but which is independent of it. Usually is a crisp partition (i.e. the data labels), and the most common approach for external validation of fuzzy partitions is to apply measures defined for crisp partitions to fuzzy partitions, using crisp partitions derived (hardened) from them. In this paper we discuss fuzzy generalizations of two well known crisp external measures, which are able to assess the quality of a partition without the hardening of . We also define a new external validity measure, that we call DNC index, useful for comparing a fuzzy to a crisp . Numerical examples based on four real world data sets are given, demonstrating the higher reliability of the DNC index.

The paper introduces the so-called coherence index of a conjunctive radial fuzzy system. The index can be treated as a measure of consistency of knowledge stored in the rule base of the system. Conjunctive fuzzy systems are the systems which employ fuzzy conjunctions for combination of antecedents (IF parts) with consequents (THEN parts) in theirs IF-THEN rules. Radial fuzzy systems are the systems which employ radial functions for representation of membership functions of incorporated fuzzy sets; and exhibiting the radial shape preservation property. Due to this property an effective mathematical analysis of these systems can be carried out.