Constructive insertion theorems and extension theorems on extremally disconnected frames
type: Item
instance of: scholarly article
title: Constructive insertion theorems and extension theorems on extremally disconnected frames
zbMATH Open document ID: 1013.06010
author: Yongming Li
author: Zhihui Li
published in: Algebra Universalis
publication date: 2003-04-23T00:00:00Z
review text: The paper is a continuation of earlier work of the first author concerning a constructive insertion theorem and an extension theorem on normal frames [\textit{Yong-Ming Li} and \textit{Guo-Jun Wang}, Commentat. Math. Univ. Carol. 38, 801-814 (1997; Zbl 0938.06008)]. For a frame \(A\), a chain indexed by rational numbers, \(C=\{ x_{\alpha }\}_{\alpha \in Q}\), is called an upper continuous chain if \(\bigvee \{ x_{\beta }\); \(\beta <\alpha \} =y_{\alpha }\) for any \(\alpha \in Q\) and \(x_{\alpha }\leq x_{\beta }\) whenever \(\alpha <\beta .\) It is called proper if \(\bigvee \{ x_{\alpha }\); \(\alpha \in Q\} =1.\) Lower continuous chains as well as proper lower continuous chains are defined dually. The authors show that the sets \(R_u(A)\) (of all upper continuous chains) or \(\overline {R_u}(A)\) (of all proper upper continuous chains) or similar sets of (proper) lower continuous chains are very useful when characterizing extremally disconnected frames (a frame \(A\) is said to be extremally disconnected if \(a'\vee a''=1\) holds for every \(a\in A,\) where \(a'\) is a pseudocomplement of \(a\)).
reviewed by: Radomír Halaš
Mathematics Subject Classification ID: 06D22
Mathematics Subject Classification ID: 54G05
Mathematics Subject Classification ID: 54B30
zbMATH DE Number: 1899966
zbMATH Keywords: frame
zbMATH Keywords: extremally disconnected frame
zbMATH Keywords: insertion theorem
zbMATH Keywords: extension theorem
MaRDI profile type: Publication
full work available at URL: s000120050186
OpenAlex ID: W2026409122
DOI: 10.1007/S000120050186
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description: scientific article; zbMATH DE number 1899966