# Formula automorphism

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## Contents

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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.