Rough weighted [formula omitted]-αβ-statistical convergence in locally solid Riesz spaces

Abstract

In this present article we set forth with the new notion of I-αβ-statistical convergence which becomes more generalized version of I-statistical convergence. Successively to compare with the following important results of Balcerzak et al. (2020) [6](i):Let I be an ideal such that I⊆Z={A⊆N:limn→∞⁡|A∩[1,n]|n=0}. Then I-statistical convergence coincides with statistical convergence,(ii):I-statistical convergence coincides with statistical convergence if I=I1n={S⊆N:∑n∈S1n<∞},(iii):Let I be a maximal ideal. Then I-statistical convergence does not coincide with statistical convergence, we produce significant results that elucidate incongruity between I-αβ-statistical convergence and I-statistical convergence. Also, following the line of works of Aytar (2008) [4], Savaş et al. (2015) [40] and Ghosal et al. (2020) [23], we propose two new notions, namely rough weighted I-αβ-statistical convergence and weighted I-αβ-statistical cluster points set over locally solid Riesz spaces. The core factor “degree of roughness” had been conventionally treated as a non-negative real number where as we invent the idea of neighborhood approximation. Finally, topological properties and relationships of the above mentioned notions are investigated.