A mathematical structure consisting of a domain together with interpretations of the symbols of a formal signature (constants, functions, relations) that makes each sentence of a given theory true in that structure.

Principle

A model provides a semantic realization of syntactic symbols: truth of sentences is evaluated by assigning meanings to symbols in a concrete structure so that the axioms of a theory hold.

Demonstration

The set G with a binary operation * and distinguished identity e that satisfies associativity, identity and inverse laws is a model of the first-order theory of groups; the integers with addition and zero form a model of the theory of abelian groups.

Misapplication

Treating any algebraic object as a model without checking the full signature or allowing partial operations (for example, treating a semigroup lacking an identity as a model of group axioms) is a misuse.

Consequence

Correct identification of models lets one transfer semantic consequences (entailments) to concrete structures, produce counterexamples to proposed theorems, and classify classes of structures by their theories.

Reversal

Instead of interpreting a theory by a structure, consider starting with a theory and asking which structures realize it; in the extreme, a theory with no models is inconsistent and its semantic perspective collapses.

Boundary

Applies to structures in a specified logical language (usually first-order); excludes informal or underspecified mathematical objects, nonstandard logics unless explicitly allowed, and partial structures when totality of operations is required by the signature.

Semantic Tension

The competing notion 'algebraic structure' emphasizes presented operations and axioms as objects, whereas 'model' emphasizes satisfaction in a semantic domain — the tension is between syntactic presentation and semantic realization.

Synthesis

A model is the semantic embodiment of a formal signature and sentences: by interpreting symbols on a domain so that the theory's sentences are true, it links syntactic axioms to concrete mathematical structures and enables semantic reasoning.