  We study the stabilized automorphism group of a subshift of finite type with
a certain gluing property called the eventual filling property, on a residually
finite group $G$. We show that the stabilized automorphism group is simply
monolithic, i.e. it has a unique minimal non-trivial normal subgroup -- the
monolith -- which is additionally simple. To describe the monolith, we
introduce gate lattices, which apply (reversible logical) gates on finite-index
subgroups of $G$. The monolith is then precisely the commutator subgroup of the
group generated by gate lattices. If the subshift and the group $G$ have some
additional properties, then the gate lattices generate a perfect group, thus
they generate the monolith. In particular, this is always the case when the
acting group is the integers. In this case we can also show that gate lattices
generate the inert part of the stabilized automorphism group. Thus we obtain
that the stabilized inert automorphism group of a one-dimensional mixing
subshift of finite type is simple.
