structure

Let $\tau$ be a signature. A $\tau$-structure $𝒜$ comprises of a set $A$, called the (or underlying set or ) of $𝒜$, 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) $f^A:A^n\to A$;

• 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 $𝒜$ is a structure, then the cardinality (or power) of $𝒜$, $|𝒜|$, is the cardinality of its $A$.

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.

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