Definable relations

An n-ary relation R on the universe M of a structure is said to be definable (or explicitly definable, or -definable) if there is a formula φ(x₁,...,x_n) such that

In other words, R is definable if and only if there is a formula φ such that

is correct.

An important special case is the definability of specific elements. An element m of M is definable in if and only if there is a formula φ(x) such that