# A generalization of the Dedekind–MacNeille completion

A generalization of the Dedekind–MacNeille completion In this paper we introduce the notion of $$Z_{\delta }$$ Z δ -continuity as a generalization of precontinuity, complete continuity and $$s_{2}$$ s 2 -continuity, where Z is a subset selection. And for each poset P, a closure space $$Z^{c}_{\delta }(P)$$ Z δ c ( P ) arises naturally. For any subset system Z, we define a new type of completion, called $$Z_{\delta }$$ Z δ -completion, extending each poset P to a Z-complete poset. The main results are: (1) if a subset system Z is subset-hereditary, then $$cl_{Z}(\Psi (P))$$ c l Z ( Ψ ( P ) ) , the Z-closure of all principal ideals $$\Psi (P)$$ Ψ ( P ) of poset P in $$Z^{c}_{\delta }(P)$$ Z δ c ( P ) , is a $$Z_{\delta }$$ Z δ -completion of P and $$Z^{c}_{\delta }(P) \cong Z^{c}_{\delta }(cl_{Z}(\Psi (P)))$$ Z δ c ( P ) ≅ Z δ c ( c l Z ( Ψ ( P ) ) ) ; (2) let Z be an HUL-system and P a $$Z_{\delta }$$ Z δ -continuous poset, then the $$Z_{\delta }$$ Z δ -completion of P is also $$Z_{\delta }$$ Z δ -continuous, and a Z-complete poset L is a $$Z_{\delta }$$ Z δ -completion of P iff P is an embedded $$Z_{\delta }$$ Z δ -basis of L; (3) the Dedekind–MacNeille completion is a special case of the $$Z_{\delta }$$ Z δ -completion. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Semigroup Forum Springer Journals

# A generalization of the Dedekind–MacNeille completion

Semigroup Forum, Volume 96 (3) – Dec 12, 2017
12 pages

/lp/springer_journal/a-generalization-of-the-dedekind-macneille-completion-6JY3zDURap
Publisher
Springer Journals
Subject
Mathematics; Algebra
ISSN
0037-1912
eISSN
1432-2137
D.O.I.
10.1007/s00233-017-9911-4
Publisher site
See Article on Publisher Site

### Abstract

In this paper we introduce the notion of $$Z_{\delta }$$ Z δ -continuity as a generalization of precontinuity, complete continuity and $$s_{2}$$ s 2 -continuity, where Z is a subset selection. And for each poset P, a closure space $$Z^{c}_{\delta }(P)$$ Z δ c ( P ) arises naturally. For any subset system Z, we define a new type of completion, called $$Z_{\delta }$$ Z δ -completion, extending each poset P to a Z-complete poset. The main results are: (1) if a subset system Z is subset-hereditary, then $$cl_{Z}(\Psi (P))$$ c l Z ( Ψ ( P ) ) , the Z-closure of all principal ideals $$\Psi (P)$$ Ψ ( P ) of poset P in $$Z^{c}_{\delta }(P)$$ Z δ c ( P ) , is a $$Z_{\delta }$$ Z δ -completion of P and $$Z^{c}_{\delta }(P) \cong Z^{c}_{\delta }(cl_{Z}(\Psi (P)))$$ Z δ c ( P ) ≅ Z δ c ( c l Z ( Ψ ( P ) ) ) ; (2) let Z be an HUL-system and P a $$Z_{\delta }$$ Z δ -continuous poset, then the $$Z_{\delta }$$ Z δ -completion of P is also $$Z_{\delta }$$ Z δ -continuous, and a Z-complete poset L is a $$Z_{\delta }$$ Z δ -completion of P iff P is an embedded $$Z_{\delta }$$ Z δ -basis of L; (3) the Dedekind–MacNeille completion is a special case of the $$Z_{\delta }$$ Z δ -completion.

### Journal

Semigroup ForumSpringer Journals

Published: Dec 12, 2017

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