# 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.
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## 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 $A$.