Abstract
Interval-valued fuzzy soft sets realize a hybrid soft computing model in a general framework. Both Molodtsov's soft sets and interval-valued fuzzy sets can be seen as special cases of interval-valued fuzzy soft sets. In this study, we first compare four different types of interval-valued fuzzy soft subsets and reveal the relations among them. Then we concentrate on investigating some nonclassical algebraic properties of interval-valued fuzzy soft sets under the soft product operations. We show that some fundamental algebraic properties including the commutative and associative laws do not hold in the conventional sense, but hold in weaker forms characterized in terms of the relation =L. We obtain a number of algebraic inequalities of interval-valued fuzzy soft sets characterized by interval-valued fuzzy soft inclusions. We also establish the weak idempotent law and the weak absorptive law of interval-valued fuzzy soft sets using interval-valued fuzzy soft J-equal relations. It is revealed that the soft product operations ∧ and ∨ of interval-valued fuzzy soft sets do not always have similar algebraic properties. Moreover, we find that only distributive inequalities described by the interval-valued fuzzy soft L-inclusions hold for interval-valued fuzzy soft sets.
Highlights
In recent years, uncertainty modelling has become an important research topic in computational intelligence and related fields
This study has been devoted to exploring some nonclassical algebraic properties of IVF soft sets under the soft product operations
Some algebraic inequalities of IVF soft sets characterized by IVF soft inclusions have been obtained and the relations among four different types of interval-valued fuzzy soft subsets have been ascertained
Summary
Uncertainty modelling has become an important research topic in computational intelligence and related fields. Algebraic operations of soft sets were initiated by Molodtsov [3] and systematically investigated by Maji et al [20]. Feng and Li [30] investigated these different types of soft subsets and the related soft equal relations in a systematic way They considered the free soft algebras associated with soft product operations. The present study aims to investigate the above line of exploration in the more general setting of interval-valued fuzzy soft sets. The last section gives conclusions to complete this study and suggests potential directions for future work
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