Closure operators defined on various sets (set of all classical fuzzy sets, set of all semi-cuts, set of all cuts in a Q-set, etc.) are investigated and it is shown how a closure operator defined on one set can be extended to a closure operator defined on another set.

This paper deals with square roots of a matrix over a complete lattice, where the matrix composition is _ U with U being an infinitely _-distributive isotonic operator. We give a general characterization for the existence of a square root of a matrix over a complete lattice. Furthermore, we give methods...

In this paper, we consider the structure of solution sets of fuzzy relation equations over complete Boolean algebras. We show that each solution of a system of fuzzy relation equations can be represented by a linear combination of a special solution of its and some certain solutions of the homogeneous...

Conditionally firing rules have been proposed by B. Moser and M. Navara in order to preserve natural properties that are not preserved by the usual setting of fuzzy inference systems such as Mamdani- Assilian rules joined to the CRI inference mechanism. In this paper, we follow this direction and show,...

Fuzzy relational compositions have been extensively studied by many authors. Especially, we would like to express studies of the fuzzy relational compositions motivated by their applications to medical diagnosis by Willis Bandler and Ladislav Kohout. These types of compositions use only two quantifiers:...

The work considers the problem of solving a system of fuzzy relational equations with inf-implication composition and introduces the concepts of a characteristic matrix and attainable components. It is first shown that solving the system is closely related with the covering problem. Further, it is proved...

The aim of this contribution is to discuss the characterizations of L-semilinear spaces which are generated by strong linearly independent vectors. First, we show that the basis in L-semilinear spaces which are generated by strong linearly independent vectors is also strong linearly independent. Then...

The law of contraposition with respect to a negation (usually strong) is one of the most studied properties in the theory of fuzzy implication functions. We already know some methods for modifying an implication with the aim that the new implication satisfies this property, these methods are called contrapositivisation....

Rule-level feature selection, also termed as rule compression, is an important technique for improving interpretability of fuzzy rule-based classifiers. In this paper we present three different rule compression algorithms and analyze their performance and characteristics on the classifiers identified...

Modus Ponens is a key property for fuzzy implication functions that are going to be used in fuzzy inference processes. In this paper it is investigated when fuzzy implication functions derived from uninorms via residuation, usually called RUimplications, satisfy the modus ponens with respect to a continuous...

This paper shows the use of Fingrams –Fuzzy Inference-grams– aimed at unveiling graphically some hidden details in the usual behavior of the precise fuzzy modeling algorithm FURIA –Fuzzy Unordered Rule Induction Algorithm–. FURIA is recognized as one of the most outstanding fuzzy rule-based classification...

We present and analyze inference method called Perception-based Logical Deduction (PbLD) aimed at the treatment of fuzzy IF-THEN rules as linguistically expressed genuine logical implications. Besides the original PbLD, we propose a new balancing variant of PbLD, introduce both variants with fuzzy inputs...

Histograms are a tool for graphical representation of frequency data and thus helpful in creating a fast understanding of, e.g., contents of frequency data. Comparing histograms is topic of increasing importance due to an increase in the availability of data sets containing frequency information. Automatic...

In this paper we introduce the concepts of a multimode multi-relational fuzzy network and a regular fuzzy equivalence on such a network, and provide procedures for computing the greatest regular fuzzy and crisp equivalences contained in a given tuple of fuzzy equivalences.

In this paper we introduce two types of simulations and five types of bisimulations for fuzzy social networks, and we study one of them – regular bisimulations. We prove that if there exists at least one regular bisimulation between two fuzzy networks, then there exists the greatest bisimulation of this...

The notion of social roles is a centerpiece of most sociological theoretical considerations. Regular equivalences arise as a result of an attempt to capture the sociological notion of a relational or structural role. Regular equivalences were introduced by White and Reitz in [28] as the least restrictive...

Recently, two new families of fuzzy implication functions called probabilistic implications and probabilistic S-implications were introduced by Grzegorzewski [6, 7, 9]. They are based on conditional copulas and make a bridge between probability theory and fuzzy logic. In this paper we generalize these...

Recently, Grzegorzewski [9] introduced two new families of fuzzy implication functions called survival implications and survival S-implications. These two classes of multivalued implications are based on conditional copulas. In the same article the author gave a motivation to his ideas and he analyzed...

We present DC* (Double Clustering with A*) as an information granulation method specifically suited for deriving interpretable knowledge from data. DC* is based on two main clustering stages: the first is devoted to compressing multi-dimensional data into few prototypes that grab the main relationships...

The purpose of these notes is to introduce the concept of a many-valued relation and to study two, in a certain sense dual, categories of such relations. Besides, we apply the concept of a manyvalued relation to the description of different approaches, both fixed-based and variable-based, to Fuzzy Topology...

Based on previous investigations, we have proposed two different methods to extend lattice-valued fuzzy connectives (t-norms, t-conorms, negations and implications) and other related operators, considering a generalized notion of sublattices. Taking into account the obtained results that we have obtained...

In the previous papers, we introduced a general principle for an introduction of new intermediate quantifiers and proved that generalized square of opposition works with them accordingly. This paper is devoted to interpretation of an intermediate quantifier “Many” and to analysis of generalized 5-square...

Paving is a method for constructing new operations from a given one. We will show that this method can be used to construct associative, commutative and monotone operations from particular given operations (from basic ‘paving stones’). We will discuss properties of the resulting operations by considering...

The disposal of many families of fuzzy implications is crucial to handle the great number of applications these operators have. In this work, the class of fuzzy (OP)-polynomial implications is introduced as those fuzzy implications satisfying the ordering property (OP) whose expression in the region...

The paper defines and studies ukasiewicz-like triangular subnorms, i.e., triangular subnorms such that their level sets are obtained as unions of the level sets of the ukasiewicz triangular norm. As a result, the paper gives a characterization of these operations.

The objective of this work is to associate the concepts of fuzzy rough sets and fuzzy topologies with fuzzy transform. It is shown here that the fuzzy transform can be viewed as fuzzy approximation operators studied in the operator-oriented view of fuzzy rough set theory and the use of fuzzy rough set...

Neuro-fuzzy networks revealed their proficiency in learning from data, while offering a transparent and somehow interpretable rule-based model. Recent research focused either on the interpretability of the chosen model or on the system performance. Regarding the interpretability, here an index to control...

Many physical theories accurately predict which events are possible and which are not, or – in situations where probabilistic (e.g., quantum) effects are important – predict the probabilities of different possible outcomes. At first glance, it may seem that this probabilistic information is all we need....

We introduce the notion of f-inclusion, which is used to describe different kinds of subsethood relations between fuzzy sets by means of monotonic functions ƒ: [0, 1] [0, 1]. We show that these monotonic functions can be considered indexes of inclusion, since the greater the function considered, the...

The notion of extensionality of a fuzzy relation w.r.t. a fuzzy equivalence was first introduced by H¨ohle and Blanchard. B elohl´avek introduced a similar definition of compatibility of a fuzzy relation w.r.t. a fuzzy equality. In [14] we generalized this notion to left compatibility, right compatibility...