## Bilinear algebras

Abbreviation: BilinA

### Definition

A bilinear algebra is a structure $\mathbf{A}=\langle A,+,-,0,\cdot,s_r\ (r\in F)\rangle$ of type $\langle 2,1,0,2,1_r\ (r\in F)\rangle$ such that

$\langle A,+,-,0,s_r\ (r\in F)\rangle$ is a vector space over a field $F$

$\cdot$ is bilinear: $x(y+z)=xy+xz$, $(x+y)z=xz+yz$, and $s_r(xy)=s_r(x)y=xs_r(y)$

It is not unusual to give several (equivalent) definitions. Ideally, one of the definitions would give an irredundant axiomatization that does not refer to other classes.

##### Morphisms

Let $\mathbf{A}$ and $\mathbf{B}$ be … . A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism: $h(x ... y)=h(x) ... h(y)$

### Definition

An is a structure $\mathbf{A}=\langle A,...\rangle$ of type $\langle ...\rangle$ such that

$...$ is …: $axiom$

$...$ is …: $axiom$

Example 1:

### Properties

Feel free to add or delete properties from this list. The list below may contain properties that are not relevant to the class that is being described.

Classtype variety

### Finite members

$\begin{array}{lr} f(1)= &1\\ f(2)= &\\ f(3)= &\\ f(4)= &\\ f(5)= &\\ \end{array}$ $\begin{array}{lr} f(6)= &\\ f(7)= &\\ f(8)= &\\ f(9)= &\\ f(10)= &\\ \end{array}$

### Subclasses

[[Lie algebras]]
[[Associative algebras]]

### Superclasses

[[Vector spaces]] reduced type