Countably complementable linear orderings
Let \({\mathbf L}\) denote the set of countable linear orderings, and \(\preccurlyeq\) the embeddability relation. Given \({\mathcal L}\in{\mathbf L}\), let \({\mathbf L}({\mathcal L})= \{{\mathcal B}\in{\mathbf L}:{\mathcal L}\npreceq{\mathcal B}\}\). A countable linear ordering \({\mathcal L}\) is said to be countably complementable if there exists a linear ordering \(\overline{\mathcal L}\), possibly uncountable, such that for every countable linear ordering \({\mathcal B}\) there holds \({\mathcal B}\in{\mathbf L}({\mathcal L})\Leftrightarrow{\mathcal B}\preceq\overline{\mathcal L}\). For \({\mathcal L}\in{\mathbf L}\) the following is defined: (1) \({\mathcal L}\) has the finitely faithful extension property if for all \({\mathcal A},{\mathcal B}\in{\mathbf L}({\mathcal L})\) there exists \({\mathcal C}\in{\mathbf L}({\mathcal L})\) such that \({\mathcal A}\preceq{\mathcal C}\) and \({\mathcal B}\preceq{\mathcal C}\). (2) \({\mathcal L}\) has the completely faithful extension property if for every set \(\{{\mathcal A}_i: i\in\omega\}\subseteq{\mathbf L}({\mathcal L})\) there exists \({\mathcal C}\in{\mathbf L}({\mathcal L})\) such that \((\forall i\in\omega)\) \({\mathcal A}_i\preceq{\mathcal C}\). (3) \({\mathcal L}\) has the totally faithful extension property if for every set \(\{{\mathcal A}_i: i\in\omega\}\subseteq{\mathbf L}({\mathcal L})\) that is totally ordered by embeddability there exists \({\mathcal C}\in{\mathbf L}({\mathcal L})\) such that \((\forall i\in\omega)\) \({\mathcal A}_i\preceq{\mathcal C}\). The main theorem of this paper gives four characterization of countable linear orderings that are countably complementable. In particular the following three properties are equivalent: \({\mathcal L}\) is countably complementable, \({\mathcal L}\) has the completely faithful extension property, \({\mathcal L}\) has the finitely faithful extension property. \textit{R. Fraïssé} [C. R. Acad. Sci., Paris 226, 1330--1331 (1948; Zbl 0034.17604)] had mentioned the following conjecture: For all countable linear orderings \({\mathcal L},{\mathcal C}_0,{\mathcal C}_1,\dots\), if for every \({\mathcal C}\) such that \(\forall n\,({\mathcal C}_n\preceq{\mathcal C})\) we have that \({\mathcal L}\preceq{\mathcal C}\), then there is some \(n\) such that \({\mathcal L}\preceq{\mathcal C}_n\). This had been shown to be false, but for the case where \({\mathcal C}_0\preceq {\mathcal C}_1\preceq{\mathcal C}_2\preceq\dots\) it is now proved, answering a question of \textit{J. G. Rosenstein} [Linear orderings. Academic Press, New York (1982; Zbl 0488.04002)]. For it is shown that every countable linear ordering has the totally faithful extension property.
scientific article; zbMATH DE number 5146176
scientific article