Complete embeddings of linear orderings and embeddings of lattice-ordered groups
For any linearly ordered set \((S,\leq)\), let \((\bar S,\leq)\) denote its Dedekind completion. The main theorem proved is: If \((C,\leq)\) is any linearly ordered set, then no non-trivial closed interval of C is dense if and only if there is a doubly homogeneous linearly ordered set \((S,\leq)\) and a complete embedding \(\Phi\) of \((C,\leq)\) into \((\bar S,\leq)\) such that \(\Phi(\bar C)\subseteq \bar S\setminus S\). Moreover, if this holds and \(k<| C|\), then (S,\(\leq)\) can be chosen of cardinality k, each \(s\in S\) has countable coterminality and whenever \(a<b\in C\) with \((a,b)=\emptyset\), \((\Phi(a),\Phi(b))\subseteq \bar S\) has countable coterminality. The proof follows the construction of Theorem 2.11 of \textit{M. Droste} and \textit{S. Shelah} [Isr. J. Math. 51, 223-261 (1985; Zbl 0608.06001)]. It is further shown that for each regular cardinal k and all regular cardinals \(\mu,\nu\leq k\), there exist up to isomorphism precisely \(2^ k\) doubly homogeneous linearly ordered sets \((S,\leq)\) of cardinality k with pairwise non-isomorphic and non- antiisomorphic Dedekind completions such that each \(s\in S\) has cofinality \(\mu\) and coinitiality \(\nu\). Moreover Aut\((C,\leq)\) can be embedded in Aut\((S;\leq)\) so as to preserve finite maxima and minima of functions. The proofs use the results of Droste and Shelah (op. cit), clever but elementary analysis, and \textit{R. M. Solovay}'s work [in Axiomatic Set Theory, Proc. Symp. Pure Math. 13 I, 397-428 (1971; Zbl 0222.02078)].
doubly homogeneous linearly ordered sets
scientific article; zbMATH DE number 4004232
scientific article