Characteristic subalgebra

Definition
Suppose $$A$$ is an algebra (in the universal algebra sense) with a given collection of algebra operations. A subalgebra $$B$$ of $$A$$ is termed a characteristic subalgebra if it satisfies the following equivalent conditions:


 * 1) For every automorphism $$\sigma$$ of $$A$$, $$\sigma(B) \subseteq B$$.
 * 2) For every automorphism $$\sigma$$ of $$A$$, $$\sigma(B)$$ is a subalgebra of $$B$$.
 * 3) For every automorphism $$\sigma$$ of $$A$$, $$\sigma(B) = B$$.

Facts
Note that the notion of subalgebra depends on precisely what operations we consider part of the algebra structure of $$A$$. For instance, if a group is treated as an algebra in the variety of groups with multiplication, identity, and inverse map operations, the subalgebras are subgroups. If, on the other hand, a group is treated as an algebra in the variety of semigroups (so we just remember the multiplication) then the subalgebras are subsemigroups.

Both the notion of "subalgebra" and the notion of "automorphism" are independent of the choice of subvariety that we are looking at, and hence the notion of "characteristic subalgebra" is also independent. For instance, the notion of characteristic subgroup remains the same whether we are working in the variety of groups or the variety of abelian groups.