𝓜𝓝-convergence and lim-inf_𝓜-convergence in partially ordered sets

Abstract In this paper, we first introduce the notion of 𝓜𝓝-convergence in posets as an unified form of O-convergence and O 2-convergence. Then, by studying the fundamental properties of 𝓜𝓝-topology which is determined by 𝓜𝓝-convergence according to the standard topological approach, an equivalent characterization to the 𝓜𝓝-convergence being topological is established. Finally, the lim-inf𝓜-convergence in posets is further investigated, and a sufficient and necessary condition for lim-inf𝓜-convergence to be topological is obtained.