Adefinable

Adefinable, also written A-definable or definable over A, is a notion in model theory describing sets or elements that can be defined by a first-order formula using parameters from a fixed set A within a given structure M. A subset D of M^n is A-definable if there exists a formula φ(x) in the language of M with parameters from A such that D = { a ∈ M^n : M ⊨ φ(a) }. The set A serves as the base of definition, and A-definability reflects how the presence of parameters enlarges the class of definable sets compared to parameter-free definability.

A-definability is said to be with respect to a chosen structure M and base set A. If A is empty, A-definable coincides with the usual notion of definability without parameters. The concept is invariant under automorphisms of M that fix A pointwise, so A-definable sets behave predictably under symmetry determined by A.

Operations and properties: The collection of A-definable sets is closed under finite unions, intersections, and complements, and under projections along A-definable maps (maps described by first-order formulas with parameters in A). The notion extends to A-definable types and A-definable families, which play a central role in stability theory and in the study of definable groups and fields.

Relation to broader concepts: Adefinability sits alongside definability over A, type-definability, and invariance notions. It provides a practical language for describing which sets and objects can be described using a fixed set of parameters, a common situation in model-theoretic investigations of structures and theories. See also definable set, parameters, model theory, and definable type.