Definition:Relational Structure

A relational structure $$\left ({S; \mathcal{R}}\right)$$ is a set $$S$$ together with a relation $$\mathcal{R}$$ on $$S$$.

Technically, $$\left ({S; \mathcal{R}}\right)$$ is an ordered pair, the first element of the pair being the set $$S$$, the second being the relation $$\mathcal{R}$$, which is itself a set of ordered pairs (being a subset of the cartesian product $$S \times S$$).

The usual (purist) approach, therefore, is to denote $$\left ({S; \mathcal{R}}\right)$$ as $$\left ({S, \mathcal{R}}\right)$$, but modern developments in notation are encouraging the semicolon; more complicated objects are clarified with this notation.