Generalised Net Convergence Structures in Posets

TL;DR

This paper introduces generalized convergence structures in posets, extending Scott and order-convergence, and characterizes when these structures are topological based on properties of the posets.

Contribution

It defines $ ext{M}$-convergence and $ ext{MN}$-convergence in posets and provides necessary and sufficient conditions for their topological nature, generalizing classical results.

Findings

Scott-convergence is topological iff the poset is continuous.

Order-convergence is topological iff the poset is $ ext{R}^*$-doubly continuous.

Generalized convergence structures unify and extend existing convergence notions.

Abstract

In this paper, we introduce the notion of $\mathcal{M}$-convergence and $\mathcal{MN}$-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 $\mathcal{R}^*$-doubly continuous.