Let $\tau$ be a signature. A $\tau$ -structure $\mathcal{A}$ comprises of a set $A$ , called the universe (or domain) of $\mathcal{A}$ , and an interpretation of the symbols of $\tau$ as follows:

• for each constant symbol $c\in\tau$ , an element $c^A\in A$ ;
• for each $n$ -ary function symbol $f\in\tau$ , a function $f^A:A^n\rightarrow A$ ;
• for each $n$ -ary relation symbol $R\in\tau$ , a subset $R^A\subseteq A^n$ .

If $\mathcal{A}$ is a structure, then the cardinality (or power) of $\mathcal{A}$ , $|\mathcal{A}|$ , is the cardinality of its universe $A$ .

If $\tau$ does not contain any function symbols, then a $\tau$ -structure is called a relational structure. If $\tau$ does not contain any relation symbols, then a $\tau$ -structure is called an algebraic structure.