Constructive insertion theorems and extension theorems on extremally disconnected frames
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 \(a\)).
scientific article; zbMATH DE number 1899966