Automorphing form

Definition
Let $$V$$ be a variety of algebras. An automorphing form for $$V$$ is an algebraic expression in $$k+1$$ variables, of which one variable is the input variable and the other $$k$$ variables are parameters, such that:

For any algebra $$A \in V$$ and any choice of values for the $$k$$ parameters, the map sending the variable to the value of the expression, is an automorphism of the algebra.

An automorphism of an algebra that arises from an automorphing form with respect to the variety is termed a form-automorphism.

Relation with other properties

 * Endomorphing form

For groups
For groups, every automorphing form essentially reduces to the following:

$$(a,x) \mapsto axa^{-1}$$

where $$a$$ is the parameter and $$x$$ is the variable.