Quantifier elimination
Related concepts

Various model-theoretic ideas are related to quantifier elimination, and there are various equivalent conditions.

Every first-order theory with quantifier elimination is model complete. Conversely, a model-complete theory, whose theory of universal consequences has the amalgamation property, has quantifier elimination.

The models of the theory of the universal consequences of a theory T {\displaystyle T} are precisely the substructures of the models of T {\displaystyle T}. The theory of linear orders does not have quantifier elimination. However the theory of its universal consequences has the amalgamation property.
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