The fuzzy concept lattice is one of the effective tools for data mining, and granular reduction is one of its significant research contents. However, little research has been done on granular reduction at different granularities in formal fuzzy contexts (FFCs). Furthermore, the complexity of the composition of the fuzzy concept lattice limits the interest in its research. Therefore, how to simplify the concept lattice structure and how to construct granular reduction methods with granularity have become urgent issues that need to be investigated. To this end, firstly, the concept of an object granule with granularity is defined. Secondly, two reduction algorithms, one based on Boolean reasoning and the other on a graph-theoretic heuristic, are formulated while keeping the structure of this object granule unchanged. Further, to simplify the structure of the fuzzy concept lattice, a partial order relation with parameters is proposed. Finally, the feasibility and effectiveness of our proposed reduction approaches are verified by data experiments.

Overlap functions, as one kind of particular binary aggregation functions, have been continuously studied in the literature for their vast preponderance in some real applications. Meanwhile, after Luccaet al.introduced the notion of preaggregation functions recently, the investigation related to such “aggregation” functions becomes an interesting, and natural research area. In this article, we continue to focus on this topic for overlap functions. First, we introduce the concept of$\vec{r}$-(quasi-)overlap functions by replacing the monotonicity of (quasi-)overlap functions with directional monotonicity, and the notions of 0-$\vec{r}$-(quasi-)overlap functions, 1-$\vec{r}$-(quasi-)overlap function, and 0,1-$\vec{r}$-(quasi-)overlap functions associated with them. And then, we propose some vital properties of$\vec{r}$-(quasi-)overlap functions. Finally, we show several construction methods of$\vec{r}$-(quasi-)overlap functions, 0-$\vec{r}$-(quasi-)overlap functions, 1-$\vec{r}$-(quasi-)overlap function, and 0,1-$\vec{r}$-(quasi-)overlap functions.

Fuzzy set theory, soft set theory and rough set theory are powerful mathematical tools for dealing with various types of uncertainty. This paper is devoted to define a broad family of soft fuzzy rough sets, each one of which, called an (I, J)-soft fu

Abstract This paper proposes a new generalization of classical real‐valued information systems, that is, interval‐valued information systems. By defining an interval‐valued dominance relation on a condition attribute, a rough set model and attribute reduction are established over interval‐valued information systems. Moreover, several interesting properties are investigated by constructive approach. Furthermore, knowledge reductions of consistent and inconsistent interval‐valued dominance decision information systems are studied, respectively. Subsequently, some descriptive theorems of knowledge reduction are presented for interval‐valued dominance decision information systems. Finally, the validity of the model and conclusions is verified by numerical example.