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The Equivalence of Two Modes of Order Convergence

Author

Listed:
  • Tao Sun

    (College of Mathematics and Physics, Hunan University of Arts and Science, Changde 415000, China)

  • Nianbai Fan

    (School of Computer Science and Engineering, Hunan University of Information Technology, Changsha 410151, China)

Abstract

It is well known that if a poset satisfies Property A and its dual form, then the o -convergence and o 2 -convergence in the poset are equivalent. In this paper, we supply an example to illustrate that a poset in which the o -convergence and o 2 -convergence are equivalent may not satisfy Property A or its dual form, and carry out some further investigations on the equivalence of the o -convergence and o 2 -convergence. By introducing the concept of the local Frink ideals (the dually local Frink ideals) and establishing the correspondence between ID-pairs and nets in a poset, we prove that the o -convergence and o 2 -convergence of nets in a poset are equivalent if and only if the poset is ID-doubly continuous. This result gives a complete solution to the problem of E.S. Wolk in two modes of order convergence, which states under what conditions for a poset the o -convergence and o 2 -convergence in the poset are equivalent.

Suggested Citation

  • Tao Sun & Nianbai Fan, 2024. "The Equivalence of Two Modes of Order Convergence," Mathematics, MDPI, vol. 12(10), pages 1-13, May.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:10:p:1438-:d:1389928
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