Abstract
The theory of rough sets is successfully applied in various algebraic systems (e.g. groups, rings, and modules). In this paper, the concept of roughness is introduced in modules of fractions with respect to its submodules. Hence, the notion of the lower and upper approximation spaces based on a submodule of the modules of fractions is introduced. Some fundamental results related to these approximation spaces are examined with examples. Moreover, this paper establishing several connections between the approximation spaces of two different modules of fractions with respect to the image and pre-image under a module homomorphism. This technique of building up a connection among the approximation spaces via module homomorphisms is useful to connect two information systems in the field of information technology.
Highlights
Data handling is encountered in many daily life problems as well as complex problems of specialized fields including computer sciences, medical sciences and environmental sciences
Fuzzy set theory is a generalization of the crisp sets
In fuzzy set theory membership function assigns the grade of membership to the elements of the universe in the unit interval [0, 1]
Summary
Data handling is encountered in many daily life problems as well as complex problems of specialized fields including computer sciences, medical sciences and environmental sciences. The work of Biswas and Nanda [3] was based only on the lower approximation Keeping this idea in mind, Kuroki and Wang [4] introduced the notion of the lower and upper approximation spaces in groups based on the normal subgroups to study the algebraic properties of rough sets in groups. Kanzanci et al [18] introduced the notion of the lower and upper approximations in quotient hypermodules with respect to fuzzy sets. The notion of the lower and upper approximations in Hv−modules is studied by Davvaz [19] He investigated the concept of rough approximation spaces of hyperrings in [20]. An equivalence relation on it’s submodules is defined and the notion of lower and upper rough approximation spaces is introduced. This is practical illustration of our work in the field of information technology
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