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.
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.
Similarly, "interpretability" may refer to a related but distinct notion about representation and provability of sentences between theories.
The partial map f from Z × Z onto Q that maps (x, y) to x/y if y ≠ 0 provides an interpretation of the field Q of rational numbers in the ring Z of integers (to be precise, the interpretation is (2, f)).
In fact, this particular interpretation is often used to define the rational numbers.
To see that it is an interpretation (without parameters), one needs to check the following preimages of definable sets in Q: