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 (the composite interpretations being viewed as operations on M and on N). Example 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: the preimage of Q is defined by the formula φ(x, y) given by ¬ (y = 0); 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(x, y) 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 = . References (Section 4.3) (Section 9.4) Model theory