Formula automorphism

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

For its use outside the wiki, please define the term when using it. If you are aware of an equivalent standard term, please leave a comment on the talk page
VIEW: Definitions built on this | Facts about this: (facts closely related to Formula automorphism, all facts related to Formula automorphism) |Survey articles about this | Survey articles about definitions built on this
VIEW RELATED: Analogues of this | Variations of this | Opposites of this |
Learn more about terminology local to the wiki | view a complete list of such terminology

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

Let $\mathcal{V}$ be a variety of algebras and $A$ be an algebra in $\mathcal{V}$. A formula automorphism is an automorphism of $A$ given by:

$x \mapsto f(x, x_2, x_3, \dots, x_n)$

where $f$ is a word (or expression) in the $x_i$s, using the operations of $\mathcal{V}$, and the $x_i$ are elements of $A$.

A strong formula automorphism is a formula automorphism whose inverse is also a formula automorphism.

Particular cases

For groups

In the case of groups, the formula automorphisms are called monomial automorphisms. An automorphism such that both that and its inverse are monomial is termed a strong monomial automorphism.

Relation with other properties

Stronger properties

• I-automorphism

Weaker properties

• Weak IC-automorphism