In model theory, interpretation of a structureM 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 ofN is stable and M is interpretable in N, then the theory of M is also stable.
Definition
An interpretation of M in Nwith parameters is a pair where n is a natural number and is a surjective map from a subset of Nn onto M such that the -preimage of every set X ⊆ Mkdefinable 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 itself is also called an interpretation. To verify that the preimage of every definable set in M is definable in N, it is sufficient to check the preimages of the following definable sets:
the graph of every function in the signature of M.
In model theory the term definable often refers to definability with parameters; if this convention is used, definability without parameters is expressed by the term 0-definable. Similarly, an interpretation with parameters may be referred to as simply an interpretation, and an interpretation without parameters as a 0-interpretation.
Bi-interpretability
If L, M and N are three structures, L is interpreted in M, and M is interpreted in N, then one can naturally construct a composite interpretation of L in N. If two structures M and N are interpreted in each other, then by combining the interpretations in two possible ways, one obtains an interpretation of each of the two structures in itself. This observation permits one to define an equivalence relation among structures, reminiscent of the homotopy equivalence among topological spaces. Two structures M and N are bi-interpretable if there exists an interpretation of M in N and an interpretation of N in M such that the composite interpretations of M in itself and of N in itself are definable in M and in N, respectively.
Example
The partial map f from Z × Z onto Q which maps to x/y if y ≠ 0 provides an interpretation of the fieldQ of rational numbers in the ringZ of integers. In fact, this particular interpretation is often used to define the rational numbers. To see that it is an interpretation, one needs to check the following preimages of definable sets in Q:
the preimage of Q is defined by the formula φ given by ¬ ;
the preimage of the diagonal of Q is defined by the formula given by = ;
the preimages of 0 and 1 are defined by the formulas φ given by x = 0 and x = y;
the preimage of the graph of addition is defined by the formula given by = ;
the preimage of the graph of multiplication is defined by the formula given by =.
