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Complex vague soft sets are essentially vague soft sets characterized by an additional parameter called the phase term which is defined over the set of complex numbers. In this study, we introduce and discuss the relations between complex vague soft sets. We present the definitions of the Cartesian product of complex vague soft sets and subsequently that of complex vague soft relations. The definition of the composition of complex vague soft sets is also provided. The notions of symmetric, transitive, reflexive and equivalence complex vague soft relations are then proposed and the algebraic properties of these concepts are verified. The relation between complex vague soft sets is then discussed in the context of a real-life problem: the relation between the financial indicators of the Chinese economy which are characterized by their degrees of influence on the financial indicators of the Malaysian economy, and the time required for the former to affect the latter. Interpretations of the results obtained from this example are then proposed by relating them to recent significant real-life events in the Chinese and Malaysian economies.
This work establishes the notion of relations between CVSSs and the composition of these relations. It improves on the concept of relations between fuzzy sets, soft sets and hybrid models of these two types of sets (such as complex fuzzy sets) because the relations are constructed based on the CVSS model which allows for an interval-based membership function that better corresponds to the intuition and subjectivity factors that exist in assigning membership values and provide an adequate parameterization tool to deal with the aspects of uncertainties and vagueness which are features that are pervasive in most real-life problems. The amplitude term of this CVS relation functions similarly to that of an ordinary membership function and describes the degree of presence or the absence of interaction between the parameters, albeit in the more accurate form of a closed interval. The phase term of this relation on the other hand, accurately captures and deals with the element of periodicity that exists in most(if not all) complex data consisting oftwo dimensional information. Verification of some of the fundamental properties of these concepts was also presented.