The main objective of this paper is the following two results. (1) There exists a computable bi-orderable group that does not have a computable bi-ordering; (2) there exists a bi-orderable, two-generated computably presented solvable group with undecidable word problem. Both of the groups can be found among two-generated solvable groups of derived length $3$.(1) [a]nswers a question posed by Downey and Kurtz; (2) answers a question posed by Bludov and Glass in Kourovka Notebook.One of the technical tools used to obtain the main results is a computational extension of an embedding theorem of B. Neumann that was studied by the author earlier. In this paper we also compliment that result and derive new corollaries that might be of independent interest.

中文翻译：

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可计算性、顺序和可解组

本文的主要目标是以下两个结果。(1) 存在一个不具有可计算双序的可计算双序群；(2) 存在一个具有不可判定词问题的二阶、二生成、可计算呈现的可解群。这两个组都可以在派生长度的两个生成的可解组中找到$3$.(1) [a]回答了唐尼和库尔茨提出的问题；(2) 回答了Bludov 和Glass 在Kourovka Notebook 中提出的问题。获得主要结果的技术工具之一是作者之前研究的B. Neumann 嵌入定理的计算扩展。在本文中，我们还对这一结果进行了补充，并得出了可能具有独立兴趣的新推论。