Convergence in [Formula: see text]-quasicontinuous posets

Area of Science:

• Order Theory
• Topology
• Theoretical Computer Science

Background:

• Nets and convergence are fundamental in topology and order theory.
• Existing generalizations of convergence often rely on join operators, limiting applicability.
• Understanding convergence in partially ordered sets (posets) is crucial for various mathematical fields.

Purpose of the Study:

• To generalize existing notions of net convergence for arbitrary posets.
• To introduce and characterize new types of continuity, specifically [Formula: see text]-continuity and [Formula: see text]-quasicontinuity.
• To establish topological characterizations for these generalized convergence and continuity concepts in posets.

Main Methods:

• Generalization of [Formula: see text]-convergence and [Formula: see text]-convergence using a cut operator.
• Development of convergence-theoretical characterizations for [Formula: see text]-continuity and [Formula: see text]-quasicontinuity.
• Investigation of the relationship between generalized convergence and topological properties in posets.

Main Results:

• A novel method for generalizing net convergence in posets is presented, replacing joins with a cut operator.
• The paper provides characterizations linking [Formula: see text]-continuity and [Formula: see text]-quasicontinuity to topological convergence.
• It is shown that a poset P is [Formula: see text]-continuous if and only if [Formula: see text]-convergence in P is topological.
• Similarly, P is [Formula: see text]-quasicontinuous if and only if [Formula: see text]-convergence in P is topological.

Conclusions:

• The cut operator provides a powerful tool for generalizing convergence in posets.
• The established equivalences between generalized continuity and topological convergence offer new insights into the structure of posets.
• These findings contribute to a deeper understanding of convergence and continuity in the context of ordered structures.