The Equivalence of Two Modes of Order Convergence

Tao Sun and Nianbai Fan ()
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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

Mathematics, 2024, vol. 12, issue 10, 1-13

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.

Keywords: order convergence; local Frink ideal (dually local Frink ideal); ID-doubly continuous poset (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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