Note on the Cantor-Bendixson rank of limit groups

We show that the Cantor-Bendixson rank of a limit group is finite as well as that of a limit group of a linear group.

On Finitely Generated Models of Theories with at Most Countably Many Nonisomorphic Finitely Generated Models

We study finitely generated models of countable theories, having at most countably many nonisomorphic finitely generated models. We intro- duce a notion of rank of finitely generated models and we prove, when T has at most countably many nonisomorphic finitely generated models, that every finitely generated model has an ordinal rank. This rank is used to give a prop- erty of finitely generated models analogue to the Hopf property of groups and also to give a necessary and sufficient condition for a finitely generated model to be prime of its complete theory. We investigate some properties of limit groups of equationally noetherian groups, in respect to their ranks.

Subgroup theorem for valuated groups and the CSA property

A valuated group with normal forms is a group with an integer-valued length function satisfying some Lyndons axioms and an additional axiom considered by Hurley. We prove a subgroup theorem for valuated groups with normal forms analogous to Grushko-Neumanns theorem. We study also the CSA property in such groups.