  Krieger's embedding theorem provides necessary and sufficient conditions for
an arbitrary subshift to embed in a given topologically mixing
$\mathbb{Z}$-subshift of finite type. For some $\mathbb{Z}^d$-subshifts of
finite type, Lightwood characterized the \emph{aperiodic} subsystems. In the
current paper we prove a new embedding theorem for a class of subshifts of
finite type over any countable abelian group. Our main theorem provides
necessary and sufficient conditions for an arbitrary subshift $X$ to embed
inside a given subshift of finite type $Y$ that satisfies a certain condition.
For the particular case of $\mathbb{Z}$-subshifts, our new theorem coincides
with Krieger's theorem. In particular, our result gives the first complete
characterization of the subsystems of the multidimensional full shift $Y=
A^{\mathbb{Z}^d}$. The natural condition on the target subshift $Y$, introduced
explicitly for the first time in the current paper, is called the map extension
property. It was introduced implicitly by Mike Boyle in the early 1980's for
$\mathbb{Z}$-subshifts, and is closely related to the notion of an absolute
retract, introduced by Borsuk in the 1930's. A $\mathbb{Z}$-subshift has the
map extension property if and only if it is a topologically mixing subshift of
finite type. Over abelian groups, a subshift has the map extension property if
and only if it is a contractible SFT as shown in work of Poirier and Salo. We
also establish a new theorem regarding lower entropy factors of
multidimensional subshifts, that extends Boyle's lower entropy factor theorem
from the one-dimensional case.
