Variety-extensible 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

Let ${\mathcal {V}}$ be a variety of algebras, and $A$ be an algebra in ${\mathcal {V}}$ . An automorphism $\sigma$ of $A$ is termed extensible' over the variety ${\mathcal {V}}$ if, whenever $A$ is embedded as a subalgebra of an algebra $B$ of ${\mathcal {V}}$ , there exists an automorphism $\varphi$ of $B$ such that the restriction of $\varphi$ to $A$ is $\sigma$ .

Particular cases

Variety of sets

In the variety of sets, every automorphism is extensible. In other words, given any set and a subset, a permutation of the subset always extends to a permutation of the whole set.

The idea here is that adding more elements to a set does not destroy the inherent symmetry between the elements already there.

Variety of groups

Further information: extensible automorphism

In the variety of groups, every inner automorphism is extensible. The extensible automorphisms conjecture states that every extensible automorphism is inner.

Relation with other properties

Stronger properties