structure

Let $\tau$ be a signature. A $\tau$-structure $\mathcal{A}$ comprises of a set $A$, called the (or underlying set or ) 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 $A$.

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.

 Title structure Canonical name Structure Date of creation 2013-05-20 18:26:21 Last modified on 2013-05-20 18:26:21 Owner CWoo (3771) Last modified by unlord (1) Numerical id 23 Author CWoo (1) Entry type Definition Classification msc 03C07 Related topic Substructure Related topic AlgebraicStructure Related topic Model Related topic RelationalSystem Defines structure Defines interpretation