Formulas and Properties for Families of Theories of Abelian Groups

First-order formulas reflect an information for semantic and syntactic properties. Links between formulas and properties define their existential and universal interrelations which produce both structural and topological possibilities for characteristics classifying families of semantic and syntactic objects. We adapt general approaches describing links between formulas and properties for families of Abelian groups and their theories defining possibilities for characteristics of formulas and properties including rank values. This adaptation is based on formulas reducing each formula to an appropriate Boolean combination of given ones defining Szmielew invariants for theories of Abelian groups. Using this basedness we describe a trichotomy of possibilities for the rank values of sentences defining neighbourhoods for the set of theories of Abelian groups: the rank can be equal -1, 0, or infinity. Thus the neighbourhoods are either finite or contain continuum many theories. Using the trichotomy we show that each sentence defining a neighbourhood either belongs to finitely many theories or it is generic. We introduce the notion of rich property and generalize the main results for these properties.