I-automorphism

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.

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

Weaker properties

 * Stronger than::Formula automorphism
 * Stronger than::Variety-extensible automorphism