Boolean matrix factorization (BMF), a popular methodology of preprocessing and analyzing 1/0 tabular data, generally handles Os and Is differently. It aims to explain Is in the data by factors, while Os are just left unexplained. This difference is mainly given by the usual data character, where 1s carry much more important information (and are much scarcer) than Os. However, in some datasets, the Is and Os are equally important. Such datasets require symmetrical handling of Is and Os. We propose a novel factorization of such data and its algorithm. Unlike usual BMF methods, factors are linearly ordered by priority in our factorization, and factors can contradict each other – meaning that one factor can put 1 where the other puts 0. In such a case, the factor with higher priority is right. We show that the proposed factorization provides a more compact data description than a straightforward application of the usual BMF methods.

We present a way to compute the set of fixpoints of a given fuzzy closure operator via algorithms for computing sets of fixpoints of ordinary closure operators. We assume that the fuzzy closure operator is given by a set of fuzzy sets generating this operator. The proposed way is based on certain reduction theorems which we provide and which relate fuzzy and ordinary closure operators and the sets of their fixpoints. We also present explicit description of selected algorithms which result using the presented approach.

We provide a new approach to fusion of Fuzzy Formal Concept Analysis and Rough Set Theory. As a starting point we take into account a couple of fuzzy relations, one of them represents the lower approximation, while the other one the upper approximation of a given data table. By defining appropriate concept-forming operators we transfer the roughness of the input data table to the roughness of corresponding formal fuzzy concepts in the sense that a formal fuzzy concept is considered as a collection of objects accompanied with two fuzzy sets of attributes—those which are shared by all the objects and those which at least one object has. In the paper we study the properties of such formal concepts and show their relationship with concepts formed by well-known isotone and antitone operators.

We present results regarding row and column spaces of matrices whose entries are elements of residuated lattices. In particular, we define the notions of a row and column space for matrices over residuated lattices, provide connections to concept lat

The present paper deals with scaling within the framework of formal concept analysis (FCA) of data with fuzzy attributes. In ordinary FCA, the input is a data table with yes/no attributes. Scaling is a process of transformation of data tables with general attributes, e.g. nominal, ordinal, etc., to data tables with yes/no attributes. This way, data tables with general attributes can be analyzed by means of FCA. We propose a new way of scaling, namely, scaling of general attributes to fuzzy attributes. After such a scaling, the data can be analyzed by means of FCA developed for data with fuzzy attributes. Compared to ordinary scaling to yes/no attributes, our scaling procedure is less sensitive to how a user defines a scale which eliminates the arbitrariness of user's definition of a scale. This is the main advantage of our approach. In addition, scaling to fuzzy attributes is appealing from the point of view of knowledge representation and is connected to Zadeh's concept of linguistic variable. We present a general definition of scaling, examples comparing our approach to ordinary scaling, and theorems which answer some naturally arising questions regarding sensitivity of FCA to the definition of a scale.