# Characteristicity is transitive for any variety of algebras

From Groupprops

ANALOGY: This is an analogue in variety of algebrass of a fact encountered in group. The old fact is: characteristicity is transitive.

View other analogues of characteristicity is transitive|View other analogues from group to variety of algebras (OR, View as a tabulated list)

## Statement

Suppose is a variety of algebras. Suppose is an algebra of , is a subalgebra of that is characteristic in (i.e., every automorphism of sends to itself) and is a subalgebra of that is characteristic in . Then, is a characteristic subalgebra of .

## Related facts

### Particular cases

### Other similar facts

- Full invariance is transitive for any variety of algebras
- Derivation-invariance is transitive for any subvariety of the variety of rings
- Derivation-invariance is transitive for Lie rings

## Proof

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