Ranks for families of regular graph theories

Abstract

This article deals with families of regular graph theories. Using invariants of regular graph theory, a criterion for e-minimality, a-minimality, and α-minimality of subfamilies of the family of all regular graph theories is obtained. These ranks and degrees play a similar role for families of theories with hierarchies for definable theories, such as Morley's Hierarchies for a fixed theory, although they have their own peculiarities. The rank of families of theories can be thought of as a measure of the complexity or richness of these families. Thus, by increasing rank by expanding families, we produce richer families and get families with infinite rank, which can be considered "rich enough". The ranks for families of the theory of regular graphs with finite and infinite diagonals are described. The family of all regular graph theories has infinite rank. This follows from the fact that if a language consists of m-ary symbols, m≥2, then the family of all theories of the given language has an infinite rank. This also means that the family of all regular graph theories is not e-totally transcendental. The results obtained can be considered as a partial answer to the question posed in [5].