Ideal lacunary statistical soft convergence of order α in soft topological spaces

Abstract

This paper introduces the concept of I-lacunary statistical soft convergence of order α for sequences of soft points in soft topological spaces. This new framework unifies ideal convergence, lacunary methods, and statistical convergence of variable order within the soft set setting. We investigate the fundamental properties of this notion, including the uniqueness of the underlying limit point in soft Hausdorff spaces and the inclusion relations between convergence classes for distinct orders satisfying 0<α≤β≤1. Furthermore, we establish a connection between lacunary and ordinary statistical soft convergence. Specifically, for the ideal of finite sets Ifin, we prove that I-lacunary statistical soft convergence of order α implies statistical soft convergence of the same order, provided the lacunary sequence satisfies a bounded ratio condition. Counterexamples are provided to demonstrate that the converse implications and general inclusions do not hold without additional assumptions.