nLab relational structure

Idea

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

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

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

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

4. an arrow structure is a relational structure proposed in modal logic to handle relational versions of groupoids;

5. a set $W$ with, on it, an equivalence relation, $\sim$, or, more generally, a family of equivalence relations, $\sim_i$, for $i$ in some indexing set.

References

See also

• Wikipedia, Relational theory

• Michael Barr, Note on a theorem of Putnam’s, Theory Appl. Categories, 3 3 (1997) 45–49 [tac:3-03]