definableks

Definableks is a term used in discussions of model theory and definability to denote elements, tuples, or substructures that are uniquely determined by a definability condition within a fixed base structure. The concept is meant to capture what it means for parts of a structure to be specified by logical formulas rather than by construction alone.

Formally, let M be a structure in a fixed first-order language L, and let Ks be a prescribed class of formulas in L (the ks-formulas) that may use parameters from M. An element a in M^n is called a definablek over M with respect to Ks if there exists a formula φ(x; p) in Ks and a parameter p from M such that M satisfies φ(a; p) and for any b in M^n, M satisfies φ(b; p) only when b = a. The collection Definableks(M; Ks) consists of all such a, taken up to automorphisms of M that fix the parameter set. This framework emphasizes definability with respect to a chosen syntactic subset of formulas.

Examples in practice depend on the chosen Ks and the ambient structure. In a field with Ks including polynomial equations with coefficients from a base set, definableks correspond to elements uniquely determined by those equations. In more complex structures, Ks might encode geometric, combinatorial, or model-theoretic data, allowing definableks to reflect richer forms of definability.

Definableks have several basic properties: they are invariant under automorphisms fixing the parameter set, and their study interacts with expansions, reducts, and notions such as elementary equivalence. They relate to broader topics in model theory, including definable sets, imaginaries, and stability theory. See also definable set and model theory for connected concepts.