Characteristicity is transitive for any variety of algebras

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 ${\mathcal {V}}$ is a variety of algebras. Suppose $A$ is an algebra of ${\mathcal {V}}$, $B$ is a subalgebra of $A$ that is characteristic in $A$ (i.e., every automorphism of $A$ sends $B$ to itself) and $C$ is a subalgebra of $B$ that is characteristic in $B$. Then, $C$ is a characteristic subalgebra of $A$.

Related facts

Particular cases

• Characteristicity is transitive
• Characteristicity is transitive for Lie rings

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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