Generalised Net Convergence Structures in Posets

Abstract

In this paper, we introduce the notion of $\mathcal{M}$-convergence and $\mathcal{M}\mathcal{N}$-convergence structures in posets, which, in some sense, generalise the well-known Scott-convergence and order-convergence structures. As results, we give a necessary and sufficient conditions for each generalised convergence structures being topological. These results then imply the following two well-established results: (1) The Scott-convergence structure in a poset $P$ is topological if and only if $P$ is continuous, and (2) The order-convergence structure in a poset $P$ is topological if and only if $P$ is ${\mathcal{R}}^{\ast }$-doubly continuous.