This is the first of two papers based on Chapter V of the author's Ph.D. thesis [K1]. The second paper [K3] will deal with a specific application of the ideas presented here to the study of modules, mostly to questions concerning the complexity of injective modules over a commutative Noetherian ring.In [D], R. Deissler introduced a “minimality rank” which I denote by “rk” here. This rank provides an ordinal measure on the difficulty of defining a given element b in a structure ℳ, allowing the use of parameters from definable subsets of ℳ. If A ⊂ M, rk(b, A, ℳ) = 0 if b is definable in ℳ by a formula with parameters from A. Roughly speaking, in the general case rk(b, A, ℳ) measures how hard we have to work at adding new parameters to A (from sets definable over A in ℳ) in order to be able to define b. “rk” is called a “minimality rank” because of the following: ℳ is a minimal model of the complete theory T = Th(ℳ) iff rk(b, Ø, ℳ) < ∞ for every b ∈ M. Deissler's rank was studied further by R. Woodrow and J. Knight [WK]. They improve on an example given by Deissler. The construction that they present illustrates well the difficulties and subtleties involved in exact calculations of rk.The central concept underlying Deissler's rank is that of a definable set. In §1 I introduce the idea of a context for definability Φ(x). A set B is Φ-definable over A if for some formula and . For Deissler's rank, Φ is the set of all formulas; for rk+ used in the study of modules, Φ+ is the set of all positive primitive formulas. Associated with each Φ is a relation ≺Φ between structures which says that Φ-definitions are preserved. In §1 I develop the basic properties of these two concepts and give a list of examples.
On commutativity of quasi-minimal groups
