Inmodel theory,interpretation of astructureM in another structureN (typically of a differentsignature) is a technical notion that approximates the idea of representingM insideN. For example, everyreduct or definitional expansion of a structureN has an interpretation inN.

Many model-theoretic properties are preserved under interpretability. For example, if thetheory ofN isstable andM is interpretable inN, then the theory ofM is also stable.

Note that in other areas ofmathematical logic, the term "interpretation" may refer to astructure,^[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.

Aninterpretation of a structureM in a structureNwith parameters (orwithout parameters, respectively)is a pair${\displaystyle (n,f)}$ wheren is a natural number and${\displaystyle f}$ is asurjectivemap from a subset ofNⁿ ontoMsuch that the${\displaystyle f}$-preimage (more precisely the${\displaystyle f^k}$-preimage) of every setX ⊆ M^kdefinable inM by afirst-order formula without parametersis definable (inN) by a first-order formula with parameters (or without parameters, respectively)^{[clarification needed]}.Since the value ofn for an interpretation${\displaystyle (n,f)}$ is often clear from context, the map${\displaystyle f}$ itself is also called an interpretation.

To verify that the preimage of every definable (without parameters) set inM is definable inN (with or without parameters), it is sufficient to check the preimages of the following definable sets:

• the domain ofM;
• thediagonal ofM²;
• every relation in the signature ofM;
• thegraph of every function in the signature ofM.

Inmodel theory the termdefinable often refers to definability with parameters; if this convention is used, definability without parameters is expressed by the term0-definable. Similarly, an interpretation with parameters may be referred to as simply an interpretation, and an interpretation without parameters as a0-interpretation.

IfL, M andN are three structures,L is interpreted inM,andM is interpreted inN, then one can naturally construct a composite interpretation ofL inN.If two structuresM andN 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 thehomotopy equivalence amongtopological spaces.

Two structuresM andN arebi-interpretable if there exists an interpretation ofM inN and an interpretation ofN inM such that the composite interpretations ofM in itself and ofN in itself are definable inM and inN, respectively (the composite interpretations being viewed as operations onM and onN).

The partial mapf fromZ × Z ontoQ that maps (x, y) tox/y ify ≠ 0 provides an interpretation of thefieldQ ofrational numbers in theringZ ofintegers (to be precise, the interpretation is (2, f)).In fact, this particular interpretation is often used todefine the rational numbers.To see that it is an interpretation (without parameters), one needs to check the following preimages of definable sets inQ:

• the preimage ofQ is defined by the formula φ(x, y) given by ¬ (y = 0);
• the preimage of the diagonal ofQ is defined by the formulaφ(x₁,y₁,x₂,y₂) given byx₁ ×y₂ =x₂ ×y₁;
• the preimages of 0 and 1 are defined by the formulas φ(x, y) given byx = 0 andx = y;
• the preimage of the graph of addition is defined by the formulaφ(x₁,y₁,x₂,y₂,x₃,y₃) given byx₁×y₂×y₃ +x₂×y₁×y₃ =x₃×y₁×y₂;
• the preimage of the graph of multiplication is defined by the formulaφ(x₁,y₁,x₂,y₂,x₃,y₃) given byx₁×x₂×y₃ =x₃×y₁×y₂.

• Ahlbrandt, Gisela; Ziegler, Martin (1986), "Quasi finitely axiomatizable totally categorical theories",Annals of Pure and Applied Logic,30:63–82,doi:10.1016/0168-0072(86)90037-0
• Hodges, Wilfrid (1997),A shorter model theory, Cambridge:Cambridge University Press,ISBN 978-0-521-58713-6 (Section 4.3)
• Poizat, Bruno (2000),A Course in Model Theory,Springer,ISBN 978-0-387-98655-5 (Section 9.4)

• Model theory
• Interpretation (philosophy)

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