structure

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. A set is a structure, with no constants, no functions, and no relations on it.
2. A partially ordered set is a structure, with one binary relation call partial order defined on the underlying set.
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. 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. 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.