Table of Contents

Nonassociative algebras

Abbreviation: JorA

Definition

A \emph{(nonassociative) 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 \emph{bilinear}: $x(y+z)=xy+xz$, $(x+y)z=xz+yz$, and $s_r(xy)=s_r(x)y=xs_r(y)$

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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: $h(x \ldots y)=h(x) \ldots h(y)$

Definition

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

$\ldots$ is …: $axiom$

$\ldots$ is …: $axiom$

Examples

Example 1:

Basic results

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.

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

[[...]] subvariety
[[...]] expansion

Superclasses

[[...]] supervariety
[[...]] subreduct

References


1) F. Lastname, \emph{Title}, Journal, \textbf{1}, 23–45 MRreview