$L$-fuzzy rough sets are extensions of the classical rough sets by relaxing theequivalence relations to $L$-relations. The topological structures induced by$L$-fuzzy rough sets have opened up the way for applications of topological factsand methods in granular computing. In this paper, we firstly prove thateach arbitrary $L$-relation can generate an Alexandrov $L$-topology.Based on this fact, w...

Mathematical theories are pervaded with the use of partial orders and more general preorders also called quasi orders i e re exive and transitive binary rela tions Preorders also play an important role in appli cations for instance as preference relations in multi criteria decision aid techniques The observation that classical relations do not allow to express partial or graded relationships ha...

Most information discovery processes need to understand the reasons of the success of the inference methods or the usability of the new information, which can sometimes be somehow explained with a few useful measures on the premises or on the chosen relations, logics and implications. This chapter presents a state of art on the latest concepts of measure, from the additive measures, to monotone...

Abstract We obtain several characterizations of the existence greatest elements a total preorder. The pertain to unconstrained preorder and constrained on every nonempty compact subset its ground set. necessary sufficient conditions are purely topological and, in case elements, formulated by making use relation set all topologies that can be defined objective relation. Observing function into t...

The unification type of an equational theory is defined using a preorder on substitutions, called the instantiation preorder, whose scope is either restricted to the variables occurring in the unification problem, or unrestricted such that all variables are considered. It is known that the unification type of an equational theory may vary, depending on which instantiation preorder is used. More...

Should testing preorder was proposed as a liveness-preserving precongruence for a process algebra except for nondeterministic choice operator [22]. However, this precongruce result cannot be generalized to other languages before carefully proved. In this paper, we show that τDes format is a precongruence format for should testing preorder. A precongruence format guarantees the precongruence of ...

In this paper, we consider the problem of solving systems of fuzzy relation equations in a space with fuzzy preorder. Two types of these systems with different compositions are considered. New solvability criteria are proposed for systems of both types. The new criteria are weaker than all the known ones that are based on the assumption that fuzzy sets on the left-hand side of a system establis...