Lstructure

L‑structure is a term used primarily in mathematical logic, especially in model theory, to denote a mathematical structure that interprets a formal language L. An L‑structure consists of a non‑empty underlying set, called the domain or universe, together with interpretations for each symbol of the language: constant symbols are assigned specific elements of the domain, function symbols are assigned actual functions on the domain of the appropriate arity, and relation symbols are assigned subsets of the appropriate Cartesian power of the domain.

Formally, if L = (C, F, R) where C is a set of constant symbols, F a set of function symbols, and R a set of relation symbols, an L‑structure M is a pair (|M|, I) where |M| is the domain and I assigns to each c ∈ C an element I(c) ∈ |M|, to each n‑ary f ∈ F a function I(f):|M|ⁿ→|M|, and to each n‑ary r ∈ R a relation I(r)⊆|M|ⁿ.

L‑structures provide the semantics for evaluating formulas of L. A sentence (a formula with no free variables) is said to be true in M, written M ⊨ φ, if the interpretation of its symbols in M makes the sentence hold according to the usual logical rules. Two L‑structures are elementarily equivalent when they satisfy exactly the same sentences of L.

Typical examples include groups, viewed as L‑structures for the language with a binary operation symbol (·), a unary inverse symbol (⁻¹), and a constant symbol for the identity; ordered fields, interpreted in the language of rings together with a binary relation symbol <; and graphs, interpreted in a language containing a single binary relation symbol for adjacency.

L‑structures are central to the study of model theory, allowing concepts such as elementary embeddings, ultraproducts, and categoricity to be defined and examined. They also appear in computer science, where structures for logical languages underpin formal verification and database theory.