Universality of the automorphism group of the real line (Q1337574)

The \(\ell\)-group (= lattice-ordered group) of all automorphisms of a linearly ordered set \(L\) is denoted by \({\mathcal A}(L)\). The group \({\mathcal A}(L)\) is said to be universal (\(\ell\)-universal) if for every linearly ordered set \(X\) with \(| {\mathcal A}(X) | \leq | {\mathcal A}(L)|\), the group \({\mathcal A}(X)\) is embeddable (\(\ell\)-embeddable) into \({\mathcal A}(L)\). The author investigates the universality and \(\ell\)- universality of the group \({\mathcal A}(\mathbb{R})\) (\(\mathbb{R}\) is the real line). The main results of the paper read as follows: 1) If \(2^{\aleph_ 0} <2^{\aleph_ 1}\), \(| {\mathcal A}(L) | \leq 2^{\aleph_ 0}\) and \({\mathcal A}(L)\) is not \(\ell\)-embeddable into \({\mathcal A}(\mathbb{R})\), then \(L\) is a Souslin line. 2) If \(2^{\aleph_ 0} = 2^{\aleph_ 1}\) or \(\lozenge^*\) holds true (for a definition of \(\lozenge^*\) see, e.g., \textit{K. Kunen}. [Set theory. An introduction to independence proofs (1980; Zbl 0443.03021)] then \({\mathcal A}(\mathbb{R})\) is not universal.