structure
Let τ be a signature.
A τ-structure
𝒜 comprises of a set A, called the (or underlying set or ) of 𝒜, and an interpretation
of the symbols of τ as follows:
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 τ contains only relation symbols, then a τ-structure is called a relational structure. If τ contains only function symbols, then a τ-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 |