Characteristicity is strongly intersection-closed for any variety of algebras

Statement

Suppose ${\mathcal {V}}$ is a variety of algebras and $A$ is an algebra in ${\mathcal {V}}$. Suppose $B_{i},i\in I$ are all characteristic subalgebras of $A$. Then, the intersection $\bigcap _{i\in I}B_{i}$ is also a characteristic subalgebra of $A$.

Particular cases

Variety	Characteristic subalgebra concept	Statement for that variety
variety of groups	characteristic subgroup	characteristicity is strongly intersection-closed
variety of Lie rings	characteristic Lie subring	characteristicity is strongly intersection-closed in Lie rings