Interpretation (model theory)

In model theory, interpretation of a structure M in another structure N (typically of a different signature) is a technical notion that approximates the idea of representing M inside N. For example, every reduct or definitional expansion of a structure N has an interpretation in N.

Many model-theoretic properties are preserved under interpretability. For example, if the theory of N is stable and M is interpretable in N, then the theory of M is also stable.

Note that in other areas of mathematical logic, the term "interpretation" may refer to a structure,^[1][2] rather than being used in the sense defined here. These two notions of "interpretation" are related but nevertheless distinct. Similarly, "interpretability" may refer to a related but distinct notion about representation and provability of sentences between theories.