Isomorphisms, definable relations, and Scott families of class 2 nilpotent groups

Abstract

The paper deals in questions on the complexity of isomorphisms and relations on universes of structures, on the number and properties of numberings in various hierarchies of sets, and on the existence of connections between semantic and syntactic properties of the structures and relations for the class of nilpotent groups of class two.We are interested in questions on the complexity of isomorphisms and relations on universes of structures, on the number and properties of numberings in different hierarchies of sets, and on the existence of connections between semantic and syntactic properties of structures and relations in known classes of algebraic systems such as groups, rings, lattices, etc. In the paper we look into the above-mentioned questions for groups in the class of nilpotent groups of class two. * Supported by the Council for Grants (under RF President) and State Aid of Fundamental Science Schools via project NSh-4413.2006.1.Pirogova 16-416, Novosibirsk 630090, Russia; tusupov@gorodok.net.

Preserving Categoricity and Complexity of Relations

“…In [11], it was shown that for α = 1, there are further examples in several familiar classes of structures, including rings and 2-step nilpotent groups. Similar example for all computable successor ordinals were constructed in [12]. In the present paper, this result is extended to computable limit ordinals.…”

“…Sufficient conditions were obtained for various properties to be transferred from a directed graph G * to a structure A * -computable categoricity, computable dimension, etc. In [12], for each computable successor ordinal α, in each of the classes above, a structure is produced that is Δ 0 α categorical but not relatively Δ 0 α categorical. The conditions guarantee that if the input graph was rigid, then the output structure is also rigid.…”

An autostable 2 nilpotent group with no Scott family of finitary formulas

“…Proposition 2 [7]. Given some countable symmetric graph A with properties (i)-(iii), there is a 2 nilpotent group A 0 of exponent p admitting an algorithm for constructing from a constructivization ν of A a constructivization μ ν of A 0 , and moreover:…”

“…Consider the various types of representation of a ∈ A 0 − Z(A 0 ) (see [7] 2. Take distinct elements x 1 and x 2 with [x 1 , x 2 ] = 1, and 0 < α, β < p.…”