Analyzing the Limit Set of Rough Ideal $\lambda\gamma$-Statistical Convergence of Order $\varrho$ in Lattice-Valued Fuzzy Normed Spaces

Analyzing the Limit Set of Rough Ideal

Abstract

This study introduces the framework of rough $\mathcal{I}$-$\lambda\gamma $-statistical convergence of order $\varrho$ within the setting of $\mathcal{L}$-fuzzy normed spaces (lattice-valued fuzzy normed spaces). This generalizes existing convergence notions by integrating ideal convergence ($\mathcal{I}$), generalized sequence transformations ($\lambda\gamma$), an arbitrary order ($\varrho$), and the concept of roughness ($r$). A primary focus is the characterization of the resulting rough limit set. We rigorously establish that, contrary to classical convergence, the limit is inherently a set. Furthermore, we prove that this limit set possesses key structural properties, specifically closure and convexity, under the topology induced by the $\mathcal{L}$-fuzzy norm. Finally, we define the corresponding notion of $\mathcal{I}$-$\lambda\gamma$-statistical cluster points of order $\varrho$ and elucidate the relationship between this set of cluster points and the rough limit set.