Let $\tau$ be a signature. A $\tau$-structure $\mathcal{A}$ comprises of a set $A$, called the universe (or underlying set 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 (or operation) $f^{A}:A^{n}\rightarrow A$;

• •

  for each $n$-ary relation symbol $R\in\tau$, a $n$-ary relation $R^{A}$ on $A$.

Some authors require that $A$ be non-empty.

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

Examples of structures abound in mathematics. Here are some of them:

1. 1.

   A set is a structure, with no constants, no functions, and no relations on it.

2. 2.

   A partially ordered set is a structure, with one binary relation call partial order defined on the underlying set.

3. 3.

   A group is a structure, with one binary operation called multiplication, one unary operation called inverse, and one constant called the multiplicative identity.

4. 4.

   A vector space is a structure, with one binary operation called addition, unary operations called scalar multiplications, one for each element of the underlying set, and one constant $0$, the additive identity.

5. 5.

   A partially ordered group is a structure like a group, but with the addition of a partial order on the underlying set.

If $\tau$ contains only relation symbols, then a $\tau$-structure is called a relational structure. If $\tau$ contains only function symbols, then a $\tau$-structure is called an algebraic structure. In the examples above, $2$ is a relation structure, while $3,4$ are algebraic structures.