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:
Some authors require that $A$ be nonempty.
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.
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  20130520 18:26:21 
Last modified on  20130520 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 