Spaces of orders and their Turing degree spectra

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• 作者：Malgorzata A. Dabkowska ; Mieczyslaw K. Dabkowski ; Valentina S. Harizanov ; Amir A. Togha
• 关键词：Orderable group ; Computable group ; Cantor set ; Turing degree ; Free group
• 刊名：Annals of Pure and Applied Logic
• 出版年：2010
• 出版时间：June 2010
• 年：2010
• 卷：161
• 期：9
• 页码：1134-1143
• 全文大小：351 K

文摘

We investigate computability theoretic and topological properties of spaces of orders on computable orderable groups. A left order on a group G is a linear order of the domain of G, which is left-invariant under the group operation. Right orders and bi-orders are defined similarly. In particular, we study groups for which the spaces of left orders are homeomorphic to the Cantor set, and their Turing degree spectra contain certain upper cones of degrees. Our approach unifies and extends Sikora’s (2004) [28] investigation of orders on groups in topology and Solomon’s (2002) [31] investigation of these orders in computable algebra. Furthermore, we establish that a computable free group F_n of rank n>1 has a bi-order in every Turing degree.