User:Paolo Liberatore~enwiki/Relational structure

In database theory, a relational structure is a mathematical representation of a database, comprising a set of relations, each associated to a relation symbol from a given set of relational symbols. Relational structures are

A language is a set whose elements are called relation symbols; each relation symbol is associated an arity. Other names used for a language are similarity type, signature, and vocabulary. Given a language comprising $$n$$ relation symbols, a relational structure over $$L$$ is a tuple $$A=(U,R_1,\ldots,R_n)$$ where $$U$$ is a set, called the universe, and $$R_1,\ldots,R_n$$ are $$n$$ relations over $$U$$; each $$R_i$$ is associated with a relation symbol of the language, and has the arity of that relation symbol.

Graphs are relational structures over a language comprising a single binary relation symbol, which represents the presence of an edge between two vertices. In other words, the language only contains a relation symbol $$E$$ of arity 2; a relational structure over this language is composed of a universe and a binary relation over the vertices; the universe is the set of nodes, and the relation contains the edges of the graph, that is, $$(a,b) \in E$$ means that nodes $$a$$ and $$b$$ are joined by an edge.

A Boolean formula in conjunctive normal form can also be seen as a relational structure over a language comprosing a unary relation symbol $$V$$ and two binary relation symbols $$P$$ and $$N$$. The universe contains the variables and the clauses; the relation associated with $$V$$ tells the variables and the clauses apart; the relations associated with $$P$$ and $$N$$ are used to represent the presence of a variable in a clause, either directly or negated.