Generalized stability of torsion-free abelian groups (Q695768)

In 1969, Shelah introduced the notion of stable theories that then initiated an intensive research on this subject. One instance is \((P,a)\)-stability, which is defined by adding to a language a unary predicate symbol and adding to types the condition of being algebraically closed for that predicate. The main theorem of the article states that any complete theory for a torsion-free abelian group is in fact \((P,a)\)-stable. The author proves this result using quantifier elimination down to positive primitive formulas and thus showing that the question of being \((P,a)\)-stable reduces to asking if a system of linear equations with integer coefficients has a solution in the algebraic closure of constants involved in the system. In the end, the author also gives an example of an abelian group that has a \((P,s)\)-unstable theory.