Generalized fuzzy orthopartitions extend the traditional concept of partitions to include both fuzziness and uncertainty. A generalized fuzzy orthopartition is a collection of intuitionistic fuzzy sets representing equivalence classes and satisfying a specific pair of axioms, which capture the idea that the classes must be disjoint and cover the initial universe. The aim of this article is twofold. Firstly, we aggregate and order generalized fuzzy orthopartitions by extending intuitionistic operations and relations. Secondly, we introduce and study entropy measures on generalized fuzzy orthopartitions by employing entropies on intuitionistic fuzzy sets already existing in the literature.

In this article, we will form a graded hexagon of opposition as an extension of the graded Peterson's square, incorporating new variations of intermediate quantifiers. We will start with the classical Aristotle's square of opposition and its extension to hexagon. Then we introduce the graded square of opposition that was suggested by Peterson and fully formalized by the authors. We also introduce quantifiers A few ( A little ) and Several and include them in the latter. Then we introduce new forms of quantifiers and study relations of contrary, sub-contrary, contradictory and sub-(sup-)altern among them. On the basis of the knowledge of the relations among all the intermediate quantifiers we construct the graded Peterson's hexagon of opposition.

Conditional relative frequency distributions are tools extensively employed in statistics and machine learning for analyzing connections of two or more categorical variables, examining patterns, and comparing data. As a first goal, we introduce the so-called conditional relative frequency distributions with undefined observations for representing frequencies characterized by uncertainty. After that, we show that conditional relative frequency distributions with undefined observations can be identified with particular generalized fuzzy orthopartitions, which are mathematical models describing vague partitions where the membership of elements to classes is only partially known.

This work focuses on fuzzy orthopartitions and credal partitions, which are distinct mathematical models representing partitions where the membership of elements to classes is only partially known. Firstly, we show that fuzzy orthopartitions and credal partitions are special cases of generalized fuzzy orthopartitions, which we introduce in this article as a new structure for modelling partitions with uncertainty. Next, we examine the connections between credal partitions and fuzzy orthopartitions, considering that both can be seen as types of fuzzy partitions (in particular, we deal with fuzzy probabilistic and Ruspini partitions). Moreover, we find that each generalized fuzzy orthopartition corresponds to a collection of zero, one, or infinitely many credal partitions; conversely, a credal partition maps to at most one generalized fuzzy orthopartition. Finally, we identify the class of all credal partitions that coincide with fuzzy orthopartitions.

Fuzzy relational formal concept analysis (FRCA)mines collections of fuzzy concept lattices fromfuzzy relational context families, which are special datasets made of fuzzy formal contexts and fuzzy relations between objects of different types. Mainly, FRCA consists of the following procedures: first, an initial fuzzy relational context family is transformed into a collection of fuzzy formal contexts; second, a fuzzy concept lattice is generated from each fuzzy formal context by using one of the techniques existing in the literature. The principal tools to transform a fuzzy context family into a set of fuzzy formal contexts are the so-calledfuzzy scaling quantifiers, which are particular fuzzy quantifiers based on the concept ofevaluative linguistic expression. FRCA can be applied whenever information needs to be extracted from multirelational datasets including vagueness, and it can be viewed as an extension of bothrelational concept analysisandfuzzy formal concept analysis. This article contributes to the development of fuzzy relational concept analysis by achieving the following goals. First of all, we present and study a new class of fuzzy quantifiers, calledt-scaling quantifiers, to extract fuzzy concepts from fuzzy relational context families. Subsequently, we provide an algorithm to generate, given a t-scaling quantifier, a collection of fuzzy concept lattices from a special fuzzy relational context family, which is composed of a pair of fuzzy formal contexts and a fuzzy relation between their objects. After that, we introduce an ordered relation on the set of all t-scaling quantifiers, which allows us to discover a correspondence among fuzzy concept lattices deriving from different t-scaling quantifiers. Finally, we discuss how the results obtained for t-scaling quantifiers can be extended to the class of fuzzy scaling quantifies. Therefore, this analysis highlights the main differences between t-scaling and fuzzy quantifiers.

We recently introduced special fuzzy quantifiers named quantifier-based operators to form extended fuzzy concept lattices and to construct graded squares, hexagons, octagons and decagons of oppositions. This article aims to extend our previous works by organizing quantifier-based operators in more general structures of oppositions: the so-called graded cubes of opposition and 5-graded cubes of opposition.

Orthopairs are, analogously to rough sets, a mathematical tool for dealing with uncertainty. Sequences of orthopairs also take into account the possibility to deal with a refinement process of information and with missing information. After recalling the main definitions, we present different operations of conjunction between sequences of orthopairs. We further present an example in which, having available non complete information about applicants for a job, two examiners evaluate them in order to find the better candidates and then their opinions are merged into an individual result.

The main, if not the only, income for social networks is from advertising. Social media platforms like Twitter have become a main stream communication medium to disseminate information and capture the interest of potential customers. So, it is crucial that the policy implemented to decide which ads to show in proximity of which user's posts, is the most profitable one: the ads shown should be as much as possible targeted to the user's interests. In this paper, we propose a context-aware advertising recommendation system that, analyzing the users' tweets during the timeline, interpretes the personal interests of users through orthopairs (they are equivalent to rough sets) to meet ads and users' interests at the right time.

We show that finite IUML-algebras, which are residuated lattices arising from an idempotent uninorm, can be interpreted as algebras of sequences of orthopairs whose main operation is defined starting from the three-valued Sobociński operator between rough sets. Our main tool is the representation of finite IUML-algebras by means of finite forests. 1

Formal Concept Analysis (FCA) and its fuzzy extension have been widely used to arrange data into a lattice that is an effective data structure useful to address several aims, such as: data mining, ontology learning and merging, and so on. In literature it is possible to distinguish two main approaches to address fuzzy FCA implementation: the one-sided threshold and the fuzzy closure one. This work focuses on a specific definition of one-sided threshold algorithm and fuzzy closure one. Specifically, it shows that these methods can be unified, since the one-sided threshold approach can be seen as a specialization of the fuzzy closure. The lattice generated using one-sided fuzzy threshold approach is a substructure of the lattice generated using the fuzzy closure approach. In addition, an experimentation has been performed on both implementations of the fuzzy FCA, one-sided threshold and fuzzy closure. In particular, the results are compared in terms of running time and number of extracted fuzzy concepts by varying the t-norm function Łukasiewicz, Gödel, and Product.