Fully invariant 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 fully invariant subalgebra or fully characteristic subalgebra if it satisfies the following equivalent conditions:


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

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.