Statistical convergence in riesz spaces

In mathematics, a Riesz space is defined as an ordered vector space in which every non-empty finite subset has a supremum and an infimum. Riesz spaces are fundamental concepts studied in functional analysis and have applications in physics and economics. There is extensive literature on the statisticalconvergence of sequences defined in Riesz spaces, as well as statistical limit points and statistical accumulation points. Statistical convergence has established a connection between statistical limit and statistical accumulation points and Riesz spaces through the properties of supremum and infimum. Statistical order convergence in Riesz spaces was first studied by Zafer Ercan in his paper "A Characterization of u-Uniformly Completeness of Riesz Spaces in Terms of Statistical u-Uniformly Pre-Completeness" for the version that holds for every u-regular Cauchy sequence. Şençimen and Pehlivan focused on the definition of statistical order convergence in Riesz spaces and the concepts of statistical monotonic convergence in their papers "Statistical Order Convergence In Riesz Space," which essentially is a special case of the definition introduced by Zafer Ercan in his a forementioned work and is more widely used in the literature. On the other hand, Abdullah Aydın, Muhammed Cınar, and Mikail Et introduced the concept of λ-statistical convergence in Riesz spaces and investigated its fundamental properties in their papers "The λ-Statistical Convergence in Riesz Spaces" using a positive scalar sequence that satisfies certain properties. Finally, Mehmet Küçükarslan and Abdullah Aydın introduced the concept of deferred statistical order convergence in Riesz spaces in their papers "Deferred statistical order convergence in Riesz spaces.". After providing basic information in this thesis, we will compile the definitions related to the statistical convergence of a sequence in Riesz spaces and introduce the fundamental results related to these topics.