Constructions of many complicated uncountable structures and Boolean algebras
A class \({\mathcal K}\) of models has the \((\alpha,\beta,\gamma,\delta)-\psi - \)bigness property iff there are \(I_ i\in {\mathcal K}\), \(i<\alpha\), \(| I_ i| =\beta\) and for different \(i,j<\alpha\), \(I_ i\) is not embeddable in \(I_ j\) in a strong sense (depending on \(\gamma\), \(\delta\), \(\psi)\). This notion is a refinement of the cardinal characteristic \(IE(\beta,T_ 1,T)\) from \textit{S. Shelah's} book ''Classification theory and the number of non-isomorphic models'' (1978; Zbl 0388.03009). The author shows that some special classes of models possess various bigness properties. Using this, many interesting Boolean algebras are constructed. A typical result is the following theorem: For every uncountable regular cardinal \(\alpha\) there is a rigid Boolean algebra B of power \(\alpha\) satisfying C.C.C. such that B has no one-to-one endomorphism different from the identity and for any a,b\(\in B\), \(0\neq a\nleq b\), \(B| a\) cannot be embedded in \(B| b.\) A Boolean algebra B is said to be \(\alpha\)-narrow if among any \(\alpha\) elements of B, two are comparable. The author shows that an \(\alpha\)- narrow Boolean algebra has a dense subset of power less than \(\alpha\) [for \(\alpha\) regular, see \textit{J. Baumgartner} and \textit{P. Komjáth}, Fundam. Math. 111, 125-133 (1981; Zbl 0452.03044)]. The existence of an endo-rigid, Bonnet-rigid indecomposable Boolean algebra of power \(\aleph_ 1\) is investigated under some additional set theoretic assumptions.
scientific article; zbMATH DE number 3878902
scientific article