Formula automorphism

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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_is, 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

Weaker properties