Definition:Relational Structure

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 $14.9$