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We introduce and study new generalizations of some rough set tools. Namely, the extended core, the generalized discernibility function, the discernibility space and the maximum partitioner. All these concepts where firstly introduced during the application of rough set theory to graphs, here we show that they have an interesting and useful interpretation also in the general setting. Indeed, among other results, we prove that reducts can be computed in incremental polynomial time, we give some conditions in order that a partition coincides with an indiscernibility partition of a given information table and we give the conditions such that a discernibility matrix corresponds to an information table.

The square of opposition is as old as logic. There has been a recent renewal of interest on this topic, due to the emergence of new structures (hexagonal and cubic) extending the square. They apply to a large variety of representation frameworks, all based on the notions of sets and relations. After a reminder about the structures of opposition, and an introduction to their gradual extensions (exemplified on fuzzy sets), the paper more particularly studies fuzzy rough sets and rough fuzzy sets in the setting of gradual structures of opposition.

An information table can change over time in several different ways: objects enter/exit the system, new attributes are considered, etc. As a consequence rough set instruments also change. At first, we recall a classification of dynamic increase of information with respect to three different factors: objects, attributes, values. Then, the corresponding changes in rough sets are discussed. Results about approximations, positive region and generalized decision are given and algorithms to update reducts and rules provided.

The term orthopair is introduced to group under a unique definition different ways used to denote the same concept. Some orthopair models dealing with uncertainty are analyzed both from a mathematical and semantical point of view, outlining similarities and differences among them. Finally, lattice operations on orthopairs are studied and a survey on algebraic structures is provided.

Different generalizations to the case of coverings of the standard approach to entropy applied to partitions of a finite universe X are explored. In the first approach any covering is represented by an identity resolution of fuzzy sets on X and a corresponding probability distribution with associated entropy is defined. A second approach is based on a probability distribution generated by the covering normalizing the standard counting measure. Finally, the extension to a generic covering of the Liang-Xu approach to entropy is investigated, both from the "global" and the "local" point of view. For each of these three possible entropies the complementary entropy (or co-entropy) is defined showing in particular that the Liang-Xu entropy is a co-entropy.

Several residuated algebras are taken into account. The set of axioms defining each structure is reduced with the aim to obtain an independent axiomatization. Further, the relationship among all the algebras is studied and their dependencies outlined. Finally, rough approximation spaces are introduced in residuated lattices with involution and their algebraic structure outlined.

Several algebraic structures (namely HW, BZMV(dM), Stonean MV and MV Delta algebras) related to many valued logical systems are considered and their equivalence is proved. Four propositional calculi whose Lindenbaum-Tarski algebra corresponds to the four equivalent algebraic structures are axiomatized and their semantical completeness is given.

A bottom-up investigation of algebraic structures corresponding to many valued logical systems is made. Particular attention is given to the unit interval as a prototypical model of these kind of structures. At the top level of our construction, Heyting Wajsberg algebras are defined and studied. The peculiarity of this algebra is the presence of two implications as primitive operators. This characteristic is helpful in the study of abstract rough approximations.

BZMV(dM) algebras are introduced as an abstract environment to describe both shadowed and fuzzy sets. This structure is endowed with two unusual complementations: a fuzzy one and an intuitionistic one similar to. Further, we show how to define in any BZMV(dM) algebra the Boolean sub-algebra of exact elements and to give a rough approximation of fuzzy elements through a pair of exact elements using an interior and an exterior mapping. Then, we introduce the weaker notion of pre-BZMV(dM) algebra. This structure still have as models fuzzy and shadowed sets but with respect to a weaker notion of intuitionistic negation similar to(alpha) with alpha is an element of [0, 1/2). In pre-BZMV(dM) algebras it is still possible to define an interior and an exterior mapping but, in this case, we have to distinguish between open and closed exact elements. Finally, we see how it is possible to define alpha-cuts and level fuzzy sets in the pre-BZMV(dM) algebraic context of fuzzy sets.