The relation that one structure can be injected into another by an injective homomorphism (embedding) that preserves the interpretations of all symbols and, depending on the precise notion, reflects relations as required. Embeddability says that a copy of the first structure exists inside the second up to isomorphism onto its image.

Principle

Structural inclusion up to isomorphism: embeddability organizes by the existence of an injective, structure-preserving map from one structure into another whose image is a substructure (or satisfies the chosen preservation/reflection conditions) of the target.

Demonstration

A simple example: the graph consisting of a path of length n is embeddable into any larger graph that contains a path of length n via the obvious injective map; the integers as an ordered group embed into the rationals as ordered groups by the inclusion map. Such embeddings witness that all relations and operations of the small structure are realized inside the larger.

Misapplication

Confusing embeddability with being a substructure (the embedded image may not be closed under additional operations in languages with function symbols unless specified), or treating non-injective homomorphisms as embeddings. Another misuse is assuming embeddability is symmetric; it is generally not.

Consequence

Embeddability induces a preorder on isomorphism types and is central to classification and universality questions; knowing which structures embed into which others informs transfer of properties, construction of universal models, and notions of minimal obstruction.

Reversal

Non-Embeddability: the negation identifies obstructions (invariants or combinatorial barriers) that prevent any injective structure-preserving map. Inverting embeddability leads to quotienting or collapsing structure, not to a symmetric relation.

Boundary

Depends on the chosen preservation/reflection convention: embeddings usually preserve functions and relations and are injective; elementary embeddings impose additional first-order preservation. Embeddability concerns maps between entire structures, not arbitrary correspondences between small parts, and does not allow adding new domain elements to the target in the embedding map.

Semantic Tension

Embeddability versus Elementary Embedding: embeddability requires preservation of signature-level structure, while elementary embedding requires preservation of all first-order formulas; some embeddings are elementary, most are not, and conflating the two misrepresents model-theoretic strength.

Synthesis

Embeddability captures when one structure can be realized inside another by an injective, structure-preserving map: it formalizes the idea of one object appearing as a faithful copy inside another and organizes comparisons, preorderings, and universality phenomena among structures.