Abstract
Abstract This study aims to explore modular inequalities of interval-valued fuzzy soft sets characterized by Jun’s soft J-inclusions. Using soft product operations of interval-valued fuzzy soft sets, we first investigate some basic properties of soft J-inclusions and soft L-inclusions. Then a new concept called upward directed interval-valued fuzzy soft sets is defined and some equivalent characterizations are presented. Furthermore, we consider modular laws in lattice theory and find that classical modular inequalities in lattice theory are not valid for interval-valued fuzzy soft sets. Finally, we present some interesting inequalities of interval-valued fuzzy soft sets by virtue of soft J-inclusions and related notions. MSC:03E72.
Highlights
It is worth noting that uncertainty arise from various domains has different nature and cannot be captured within a single mathematical framework
3 Some basic properties of IVF soft inclusions Here we propose several basic inequalities of IVF soft sets characterized by IVF soft inclusions, which are useful in subsequent discussions
4 Upward directed IVF soft sets and idempotency we investigate algebraic properties of soft product operations of IVF soft sets by considering idempotency
Summary
It is worth noting that uncertainty arise from various domains has different nature and cannot be captured within a single mathematical framework. In addition to probability theory and statistics, we currently have some advanced soft computing methods such as fuzzy sets [ ], rough sets [ ], and soft sets [ ]. Molodtsov’s soft set theory provides a relatively new mathematical approach to dealing with uncertainty from a parameterization point of view. Some researchers endeavored to enrich soft sets by combining them with other soft computing models such as rough sets and fuzzy sets. Maji et al [ ] initiated the study on hybrid structures involving both fuzzy sets and soft sets. They introduced the notion of fuzzy soft sets, which can be seen as a fuzzy generalization of Molodtsov’s soft sets. Feng and Li [ ] investigated different types of soft subsets and the related soft equal relations in a systematic way
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