Relational structures are models for a relational theory, that is, a logical theory whose signature is relational.

Definition

A relational structure is a tuple $\mathfrak{F}$ whose first component $W$, is a non-empty set, and whose remaining components are relations on $W$. (We assume that there is at least one relation given.)

In their use in the Kripke semantics of modal logics, the set $W$ is sometimes called the universe (perhaps better the domain) and the elements of $W$ are called ‘worlds’, amongst a host of other names! This leads to the terminology ‘possible world semantics’ which is sometimes used.

Examples

poset, $(W;\leq)$ or more generally a set, $W$, with a family of partial orders, $\{\leq_i\}$, on it;

transition system, $(S; \{R_e\mid e\in E\})$, (see discussion on the transition system’s page);

a rooted tree can be considered as a relational structure on its set of nodes, by specifying properties of a successor relation;