An algebraic or logical property of a class of (finite) structures stating that every finite partial substructure (or finite partial algebra satisfying the class' diagram locally) can be embedded into some finite full structure in the class; informally, finite consistent local pieces extend to finite global members of the class.

Principle

The organising idea is a local-to-global finite extension: finite partial consistency (no local contradiction among finitely many elements and relations) should be realizable inside a finite model of the whole class, so local constraints do not force infinite completion.

Demonstration

Example: certain varieties of finite algebras and some classes of relational structures have FEP: given a finite partial multiplication table consistent with a Boolean algebra's laws one can embed that partial table into a finite Boolean algebra, showing the partial structure extends to a finite full structure in the class.

Misapplication

Inferring from FEP that the whole class is finitely axiomatizable, decidable, or closed under arbitrary constructions; FEP concerns only embeddability of finite partial structures and does not automatically yield global algebraic or algorithmic properties.

Consequence

When a class has FEP one can often reduce checking of finite satisfiability or local constraints to finite model search, derive finite model properties for universal theories, and obtain transfer results useful in decidability and completeness arguments for finite models.

Reversal

Failure of FEP means there are finite partial structures consistent with the defining relations that cannot be embedded into any finite full structure of the class, so finite local consistency does not guarantee a finite completion and finite-model reasoning is obstructed.

Boundary

Applies to finite partial substructures relative to a specified signature and class; it excludes statements about infinite partial structures, about embeddings into infinite models, and about syntactic derivability outside the algebraic or relational setting specified.

Semantic Tension

There is tension between FEP and compactness or closure properties: FEP is a finitary extension principle that can conflict with infinitary constructions, and it trades off general model-theoretic closure for stronger finite-model control.

Synthesis

FEP formalises the idea that locally consistent finite pieces of structure can be realised inside finite global models of the class: it is a pragmatic bridge from finite local specifications to finite realizations, crucial in algebraic model theory and finite-model decidability.