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In this paper, the concepts of set-valued anti-homomorphism and strong set-valued anti-homomorphism of $\Gamma$-semigroup are introduced. The notions of generalized lower and upper approximation operators, constructed by means of set-valued mapping, which is a generalization of the notion of lower and upper approximation of $\Gamma$-semigroup, are provided. Some significant properties of lower and upper approximations of $\Gamma$-semigroup under (strong)set-valued anti-homomorphisms are discussed. We provide examples for the properties of lower and upper approximations of $\Gamma$-semigroup under (strong)set-valued anti-homomorphisms. This article explores the notion of a set-valued anti-homomorphism for $\Gamma$-semigroups, which is a generalization of anti-homomorphism in general, Comment: 23 pages

Rough graph is the graphical structure of information system with imprecise knowledge. Tong He designed the properties of rough graph in 2006[6] and following that He and Shi introduced the notion of edge rough graph[7]. He et al developed the concept of weighted rough graph with weighted attributes[6]. In this paper, we introduce a new type of labeling called Even vertex {\zeta}- graceful labeling as weight value for edges. We investigate this labeling for some special graphs like rough path graph, rough cycle graph, rough comb graph, rough ladder graph and rough star graph., Comment: 10 pages, 13 figures

Rough membership function defines the measurement of relationship between conditional and decision attribute from an Information system. In this paper we propose a new method to construct rough graph through rough membership function $\omega_{G}^F(f)$. Rough graph identifies the pattern between the objects with imprecise and uncertain information. We explore the operations and properties of rough graph in various stages of its structure., Comment: 13 pages, 11 figures

In this paper, we prove the lower bound for the number of balancing non-Wieferich primes in arithmetic progressions. More precisely, for any given integer $r\geq2$ there are $\gg\log x$ balancing non-Wieferich primes $p\leq x$ such that $p\equiv\pm1 \pmod{r}$, under the assumption of the $abc$ conjecture for the number field $\mathbb{Q}(\sqrt{2})$. Further, we discuss some applications of balancing sequences in cryptography., Comment: 18 pages, some applications of balancing sequences are given

We prove the lower bound for the number of Lucas non-Wieferich primes in arithmetic progressions. More precisely, for any given integer $k\geq 2$ there are $\gg \log x$ Lucas non-Wieferich primes $p\leq x$ such that $p\equiv\pm1\pmod{k}$, assuming the $abc$ conjecture for number fields. Further, we discuss some applications of Lucas sequences in Cryptography., Comment: 12 pages, added some applications in cryptography

We consider the negative polynomial Pell's equation $P^2(X)-D(X)Q^2(X)=-1$, where $D(X)\in \mathbb{Z}[X]$ be some fixed, monic, square-free, even degree polynomials. In this paper, we investigate the existence of polynomial solutions $P(X), \, Q(X)$ with integer coefficients., Comment: 10 pages, some more equations added