# Relational structures

## 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 $𝔉$ 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, $\left(W;\le \right)$ or more generally a set, $W$, with a family of partial orders, $\left\{{\le }_{i}\right\}$, on it;

2. transition system, $\left(S;\left\{{R}_{e}\mid e\in E\right\}\right)$, (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 $i$ is some indexing set.

Revised on December 24, 2010 07:52:16 by Toby Bartels (75.88.75.53)