singledefinable

Singledefinable is a term used in mathematical logic to describe a subset, relation, or element of a structure that can be defined by a single first-order formula in a given language, possibly with parameters from the ambient structure. Formally, let M be a structure in language L. A subset S of the domain of M is singledefinable if there exists a formula φ(x) (with or without parameters from M) such that S = {a in M : M satisfies φ(a)}. If no parameters are allowed, the subset is called parameter-free definable; if parameters are allowed, it is parameter-definable.

Examples are drawn from familiar contexts. In the real field R with the usual operations, the set of nonnegative numbers is singledefinable by the formula ∃y (x = y^2). In a group G, the set of elements of order dividing 2 is definable by φ(x) = x^2 = e. These illustrate how a single logical condition can pick out precisely the elements of interest.

Relation to broader definability: singledefinable is a specialization of definability, emphasizing that one formula suffices to describe the set, relation, or element. In many treatments, the term is not standard, and authors simply say that a set is definable (or parameter-definable) by a single formula. The phrase helps distinguish between definable objects defined by one formula versus those requiring a family of formulas or more complex schemas.

See also: definable set, parameter-definable, parameter-free definable, first-order logic.