I-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.
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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.

Particular cases

For groups

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

Relation with other properties

Weaker properties