I-automorphism

This article defines a property that can be evaluated for an automorphism of an algebra in a variety of algebras. The evaluation of that property depends on the ambient variety, and not just on the automorphism or the algebra.
View all such properties

Definition

Suppose ${\mathcal {V}}$ is a variety of algebras, and $A$ is an algebra in ${\mathcal {V}}$. An I-automorphism of $A$ is an automorphism that can be expressed as:

$x\mapsto \varphi (x,u_{1},u_{2},\dots ,u_{n})$

where $u_{1},u_{2},\dots ,u_{n}\in A$ are fixed, and $\varphi$ is a word in terms of the operations of the algebra,with the property that for any algebra $B$ of ${\mathcal {V}}$, and any choice of values $v_{1},v_{2},\dots ,v_{n}\in B$, the map:

$x\mapsto \varphi (x,v_{1},v_{2},\dots ,v_{n})$

gives an automorphism of $B$.

In other words $\varphi$ is guaranteed to give an automorphism.

Particular cases

For groups

In the variety of groups, the I-automorphisms are precisely the inner automorphisms: the automorphisms of the form $x\mapsto gxg^{-1}$. For full proof, refer: Inner automorphisms are I-automorphisms in variety of groups

Relation with other properties

Weaker properties

• Formula automorphism
• Variety-extensible automorphism