Generalized ideal convergence on quasi-continuous domains

TL;DR

This paper introduces generalized ideal convergence concepts in quasi-continuous domains, exploring their relations with topologies like Scott and Lawson, and establishing conditions for topological convergence in directed complete posets.

Contribution

It defines generalized ideal inf-limit and final lower bound limit, analyzing their topological properties and relations with existing domain topologies, especially in quasi-continuous and continuous posets.

Findings

Generalized ideal inf-limit topology aligns with Scott topology in directed complete posets.

Convergence is topological iff the poset is quasi-continuous.

Generalized ideal final lower bound limit topology aligns with Lawson topology in quasi-continuous domains.

Abstract

In this paper,the concepts of generalized ideal inf-limit and generalized ideal final lower bound limit are introduced in the directed complete poset,and their relations with Scott topology and Lawson topology are studied. The main results are as follows: (1) On directed complete posets,generalized ideal inf-limit topology is consistent with Scott topology; (2) Generalized ideal inf-limiti convergence is topological if and only if directed complete posets are quasi-continuous domains; (3) In quasi-continuous domain,generalized ideal final lower bound limit topology is consistent with Lawson topology;(4) In meet continuous directed complete posets,the generalized ideal final lower bound limit convergence is topological if and only if the directed complete poset is continuous.