Structure (mathematical logic)

Mapping of mathematical formulas to a particular meaning, in universal algebra and in model theory

The ordinary signature for set theory includes a single binary relation ∈. A structure for this signature consists of a set of elements and an interpretation of the ∈ relation as a binary relation on these elements.

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

By Beth's theorem, every implicitly definable relation is explicitly definable.

Many-sorted structures are often used as a convenient tool even when they could be avoided with a little effort. But they are rarely defined in a rigorous way, because it is straightforward and tedious (hence unrewarding) to carry out the generalization explicitly.