# Definition:Relational Structure

## Contents

## Definition

A **relational structure** is an ordered pair $\left ({S, \mathcal R}\right)$, where:

- $S$ is a set
- $\mathcal R$ is an endorelation on $S$.

## Also known as

A relational structure may also be called a **relational system**.

## Remarks

In the context of class-set theory, it is common to abuse notation by writing $\left({C, \mathcal R}\right)$ when $C$ is a class and $\mathcal R$ is a relation on $C$, and to call this a relational structure.

One must take care, as such a relational structure is not actually an ordered pair, or even, in most theories, a mathematical object of any kind, but only notational shorthand for a concept.

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Exercise $14.9$ - 2010: Raymond M. Smullyan and Melvin Fitting:
*Set Theory and the Continuum Problem*(revised ed.): $\S 10.1$