Computable Ordered Abelian Groups and Fields

Abstract. We present transformations of linearly ordered sets into ordered abelian groups and ordered fields. We study effective properties of the transformations. In particular, we show that a linear order L has a 0 2 copy if and only if the corresponding ordered group (ordered field) has a computable copy. We apply these codings to study the effective categoricity of linear ordered groups and fields. Key words: computable algebra, effective categoricity. We study complexity of isomorphisms between computable copies of ordered abelian groups and fields1 . Recall that an ordered abelian group is one in which the order is compatible with the additive group operation. Ordered fields are defined in a similar manner. We say that an ordered abelian group A = (A; +, ) is computable if its domain A, the operation +, and the relation are computable. Similarly, a field is computable if its domain and its basic operations are computable. If A is computable and isomorphic to B, we say that A is ...

• Type: Conference
• Year: 2010
• Where: CIE
• Authors: Alexander G. Melnikov
• Keywords: Abelian Groups, Applied Computing, CIE 2010, Computable, Ordered Abelian Group