A bijection between finite substructures of two structures (or between finite subsets with induced structure) that preserves all function and relation interpretations on its domain and image. A partial isomorphism need not be defined on the whole domain; it witnesses local structural agreement between models.

Principle

Local bijective preservation: the organizing constraint is that on the finite domain where the map is defined, every relation and function symbol has the same truth-value and value after mapping, so the map is an isomorphism between the induced finite substructures.

Demonstration

In Ehrenfeucht–Fraïssé games, a position is given by a finite partial isomorphism between two structures: the current bijection matches selected elements so that tuples satisfy the same atomic formulas. Example: between two graphs, a bijection between finite vertex sets that preserves adjacency and non-adjacency is a finite partial isomorphism.

Misapplication

Assuming a finite partial isomorphism extends to a full isomorphism without further argument, or treating any partial homomorphism (not injective or not relation-reflecting) as a partial isomorphism. Another misuse is conflating 'partial isomorphism' with 'isomorphism on a substructure' when domain/image are not closed under functions in the language.

Consequence

Partial isomorphisms enable local-to-global arguments (via back-and-forth) to establish elementary equivalence or full isomorphism for countable structures; they formalize the notion of two structures being indistinguishable by formulas of bounded quantifier depth.

Reversal

Total Isomorphism: the reversal is a global bijection preserving all structure everywhere. Failing to extend partial isomorphisms points to inherent differences obstructing elementary equivalence or isomorphism.

Boundary

Applies to bijections between finite induced pieces (or finite partial maps) that exactly preserve interpretations; it excludes arbitrary partial functions, non-injective maps, and maps that only preserve positive information without reflecting it. Scope often restricts to finite domains when used in back-and-forth arguments.

Semantic Tension

Partial Isomorphism versus Partial Homomorphism: a partial isomorphism is bijective and reflects as well as preserves atomic facts, whereas a partial homomorphism need only preserve positive relations and need not be injective; conflation of the two weakens arguments relying on bijectivity or reflection.

Synthesis

A partial isomorphism is a finite, bijective agreement between induced substructures: it captures precise local indistinguishability by requiring exact preservation and reflection of the language on a finite domain, and it serves as the elementary building block for back-and-forth constructions and local comparison of models.