Universal sequences for the order-automorphisms of the rationals (Q2822137)

Let \(A^{+}\) denote the semigroup freely generated by a finite alphabet \(A\) of size \(m\) and let \(T\) be a subset of a semigroup \(S\). A sequence \(w_{1} ,w_{2},\dots\) of words in \(A^{+}\) is said to be \(m\)-letter universal for \(T\) as a subset of \(S\) if for each sequence \(t_{1},t_{2},\dots\) of elements from \(T\;\)there exists a homomorphism \(A^{+}\rightarrow S\) which maps \(w_{i}\longmapsto t_{i}\) for each positive integer \(i\). Let \(\mathrm{Aut}(\mathbb{Q} ,\leq)\) be the group of all order-preserving permutations of the rationals. In [C. R., Math., Acad. Sci. Paris 342, No. 6, 377--380 (2006; Zbl 1107.20305)], \textit{A. Khelif} announced without proof that there exists a finite \(m\) such that \(\mathrm{Aut}(\mathbb{Q} ,\leq)\) is \(m\)-letter universal. The main result of the present paper is a proof that \(\mathrm{Aut}(\mathbb{Q},\leq)\) is \(2\)-letter universal. In particular, this shows that every countable subset of \(\mathrm{Aut}(\mathbb{Q},\leq)\) is contained in a subsemigroup generated by two elements of the group.NEWLINENEWLINEA group \(G\) has Bergman's property if for each generating set \(X\) of \(G\) with \(X=X^{-1}\) and \(1\in X\) there exist a positive integer \(N\) such that \(G=X^{N}\) (see [\textit{G. M. Bergman}, Bull. Lond. Math. Soc. 38, No. 3, 429--440 (2006; Zbl 1103.20003)]). Known results show that if \(G\) is not finitely generated and is \(m\)-letter universal for some \(m\) then \(G\) has strong cofinality \(\operatorname{scg}(G)>\aleph_{0}\); the main theorem of this paper then implies that \(\mathrm{Aut}(\mathbb{Q},\leq)\) has Bergman's property (see [\textit{C. Gourion}, C. R. Acad. Sci., Paris, Sér. I 315, No. 13, 1329--1331 (1992; Zbl 0797.20002); \textit{F. Galvin}, J. Lond. Math. Soc., II. Ser. 51, No. 2, 230--242 (1995; Zbl 0837.20005); \textit{M. Droste} and \textit{W. C. Holland}, Forum Math. 17, No. 4, 699--710 (2005; Zbl 1093.20016)]).