Ordering coproducts of groups and semigroups
\textit{A. A. Vinogradov} [Mat. Sb., Nov. Ser. 25(67), 163-168 (1949; Zbl 0038.159)] proved that a coproduct of orderable groups is orderable, and \textit{R. E. Johnson} [Proc. Am. Math. Soc. 19, 697-700 (1968; Zbl 0157.050)] extended this result to semigroups. A simpler version of their proof is given, using \(2\times 2\) matrices over a polynomial ring in place of infinite matrices, and the same method is used to study orderability of coproducts of groups with amalgamation. In particular, it is proved that the coproduct of two or more copies of an orderable group H with amalgamation of a subgroup \(T\subseteq H\) is orderable if and only if (a) T is the equalizer of a pair of homomorphisms from H into another orderable group, equivalently, (b) within some orderable overgroup of H, T can be written as the intersection of H with the centralizer of some subset, equivalently, (c) T is convex under some one-sided group-ordering of H. For coproducts of families of not necessarily isomorphic overgroups of a common group T, some sufficient conditions and some necessary conditions for orderability are obtained, and various counterexamples given. The last section treats one-sided orderability. It is deduced from the Kurosh subgroup theorem that the coproduct of a family of right orderable groups \(H_ i\) admit embeddings in some right orderable group by homomorphisms agreeing on T. In particular, a coproduct of copies of a single right orderable group H with amalgamation of any subgroup T is right orderable. Finally, the matrix method of the earlier sections is adapted to show that a coproduct (with only \(\{\) \(1\}\) amalgamated) of right orderable cancellative semigroups with 1 is right orderable.
scientific article; zbMATH DE number 4089622
scientific article