High Quality Content by WIKIPEDIA articles! In model theory, interpretation of a structure M in another structure N 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. An interpretation of M in N with parameters is a pair where n is a natural number and f is a surjective map from a subset of Nn onto M such that the f-preimage of every set X Mk definable in M by a first-order formula without parameters is definable by a first-order formula with parameters. Since the value of n for an interpretation is often clear from context, the map f itself is also called an interpretation

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