Granular computing in zero-divisor graphs of Zn

Abstract

In this article, we study the zero-divisor graphs $\Gamma \left(Z_n\right)$ of rings of integers modulo $n$ as information systems $I\left(\Gamma \left(Z_n\right)\right)$ using equivalence classes and rough sets. Equivalence classes are referred as granules and partitions are referred as indiscernible partitions. We define an indiscernibility relation on the vertex set and identify different sets of attributes that induce the same indiscernibility partition. A reduct is a minimal subset of attributes which yields the same partition as the original attribute set. We compute all reducts of the defined information system and classify them in to two types including: (i) the set $P$ of all prime divisors of $n$ and (ii) the set consisting of $P\setminus p_i$, prime powers of $p_i$ in the prime factorization of $n$ and the elements of the form $p_i{p}_j$, where $p_j\in P\setminus p_i$. Moreover, we give the structures and the cardinalities of the attribute subsets whose removal yields a different indiscernibility partition than that of the set of all attributes, referred as essential sets. We prove that the essential sets of $I\left(\Gamma \left(Z_n\right)\right)$ either consist of two prime divisors of $n$ or one prime divisor $p_i$ combined with prime powers of $p_i$. Further, we determine the lower and upper approximations of various vertex subsets to study the properties of the zero-divisor graphs. We also study properties of the rough membership function for $\Gamma \left(Z_n\right)$. Furthermore, we introduce the class-based discernibility matrix induced by indiscernibility classes of zero divisors and determine general form of its entries. We also prove that the minimal entries of the class-based discernibility matrix coincide with the essential sets of $\Gamma \left(Z_n\right)$. Based on these results, we determine information-granularity measures corresponding to the notable partitions of $\Gamma \left(Z_n\right)$ and provide an example to establish consistency of the proved results with these well-known measures. Thus, starting from introducing an information system, we investigate the components of granular computing and utilize our findings to compute information granularity measures, contributing to a deeper understanding of the zero divisor graphs via rough set theory.