Structure embedding ($f:\mathcal{M}\hookrightarrow \mathcal{N}$)

A structure embedding is an injective map respecting constants and operations and preserving and reflecting relations. Equivalently it preserves all atomic formulas and their negations. It identifies the source with a substructure of the target. An elementary embedding additionally preserves every first-order formula.

A structure isomorphism is a bijective structure embedding: it preserves constants and functions, and both preserves and reflects relations. Its inverse has the same properties. It therefore preserves truth of every first-order formula, by structural induction, so it preserves realization and omission of parameter-free types.