Basic model theoretic notions including languages, signatures, substructures, embeddings, elementary equivalence, isomorphism, and examples.
Model theory studies mathematical structures through the formal languages used to describe them, and it asks how much information about a structure can be expressed by first order formulas.
A structure consists of a domain together with interpretations of the symbols in a language, and this framework allows groups, rings, fields, ordered sets, graphs, and many other mathematical objects to be treated within one common logical setting.
The chapter begins with languages and signatures, where the non logical symbols of a theory are specified, including constant symbols, function symbols, and relation symbols, each with its assigned arity.
Substructures and embeddings are then introduced to compare structures that share the same language, and these notions make precise when one structure sits inside another while preserving the interpretation of the relevant symbols.
The chapter next studies elementary equivalence, which captures the idea that two structures satisfy exactly the same first order sentences, even if they are not isomorphic as mathematical objects.
Isomorphism and invariants are then discussed as stronger ways of comparing structures, where an isomorphism preserves the entire formal structure and allows two models to be regarded as the same from the viewpoint of the language.
The final part of the chapter gives examples from algebra and geometry, showing how familiar mathematical objects can be represented as first order structures and how logical notions reveal similarities and differences between them.
These ideas form the basic vocabulary of model theory, where formulas, structures, and maps between structures are studied together as a unified mathematical system.