 Elementary embedding

An elementary embedding $j:M\to N$ between first-order structures preserves and reflects every first-order formula with parameters: for every finite tuple $\bar{a}$ from $M$,In particular it is injective, by applying the condition to equality, and its image is an elementary substructure of $N$. An elementary inclusion is the defining map of an elementary extension.

For example, the rational order is an elementary substructure of the real order in the language containing only $<$, by quantifier elimination for dense linear orders without endpoints. Elementarity is stronger than preserving just the atomic relations. For embeddings between suitable transitive models of set theory, one can additionally ask which ordinals are moved; the least moved ordinal, when there is one, is the critical point of an elementary embedding. This ordinal notion is not part of the definition for arbitrary structures.