# Definition:Relational Structure

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## Definition

A **relational structure** is an ordered pair $\struct {S, \RR}$, where:

- $S$ is a set
- $\RR$ is an endorelation on $S$.

## Also known as

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

## Warning

In the context of class theory, it is common to abuse notation by writing $\struct {C, \RR}$ when $C$ is a class and $\RR$ is a relation on $C$, and to call this a **relational structure**.

One must take care, as if $C$ is a proper class then it cannot be a member of any class.

By the set-theoretic definitions for ordered pairs, if $\struct {C, \RR}$ is an ordered pair then $C$ is a member of some class, which is a contradiction.

Thus, $\struct {C, \RR}$ is not a formal mathematical object of any kind, let alone an ordered pair, but only notational shorthand for a concept.

## Also see

- Results about
**relational structures**can be found here.

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 14$: Orderings: Exercise $14.9$