##  [Partial Isomorphism](/partial-isomorphism-0) 

 Definition

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

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

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

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

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

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

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

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

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.