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- Title
A generalization of the Dedekind-MacNeille completion.
- Authors
Zhang, Zhongxi; Li, Qingguo
- Abstract
In this paper we introduce the notion of Zδ<inline-graphic></inline-graphic>-continuity as a generalization of precontinuity, complete continuity and s2<inline-graphic></inline-graphic>-continuity, where Z is a subset selection. And for each poset P, a closure space Zδc(P)<inline-graphic></inline-graphic> arises naturally. For any subset system Z, we define a new type of completion, called Zδ<inline-graphic></inline-graphic>-completion, extending each poset P to a Z-complete poset. The main results are: (1) if a subset system Z is subset-hereditary, then clZ(Ψ(P))<inline-graphic></inline-graphic>, the Z-closure of all principal ideals Ψ(P)<inline-graphic></inline-graphic> of poset P in Zδc(P)<inline-graphic></inline-graphic>, is a Zδ<inline-graphic></inline-graphic>-completion of P and Zδc(P)≅Zδc(clZ(Ψ(P)))<inline-graphic></inline-graphic>; (2) let Z be an HUL-system and P a Zδ<inline-graphic></inline-graphic>-continuous poset, then the Zδ<inline-graphic></inline-graphic>-completion of P is also Zδ<inline-graphic></inline-graphic>-continuous, and a Z-complete poset L is a Zδ<inline-graphic></inline-graphic>-completion of P iff P is an embedded Zδ<inline-graphic></inline-graphic>-basis of L; (3) the Dedekind-MacNeille completion is a special case of the Zδ<inline-graphic></inline-graphic>-completion.
- Subjects
GENERALIZATION; POLYNOMIALS; MATHEMATICS theorems; NUMERICAL analysis; ALGORITHMS
- Publication
Semigroup Forum, 2018, Vol 96, Issue 3, p553
- ISSN
0037-1912
- Publication type
Article
- DOI
10.1007/s00233-017-9911-4