  We study the ground-state space properties for frustration-free Hamiltonians.
We introduce a concept of `reduced spaces' to characterize local structures of
ground-state spaces. For a many-body system, we characterize mathematical
structures for the set $\Theta_k$ of all the $k$-particle reduced spaces, which
with a binary operation called join forms a semilattice that can be interpreted
as an abstract convex structure. The smallest nonzero elements in $\Theta_k$,
called atoms, are analogs of extreme points. We study the properties of atoms
in $\Theta_k$ and discuss its relationship with ground states of $k$-local
frustration-free Hamiltonians. For spin-1/2 systems, we show that all the atoms
in $\Theta_2$ are unique ground states of some 2-local frustration-free
Hamiltonians. Moreover, we show that the elements in $\Theta_k$ may not be the
join of atoms, indicating a richer structure for $\Theta_k$ beyond the convex
structure. Our study of $\Theta_k$ deepens the understanding of ground-state
space properties for frustration-free Hamiltonians, from a new angle of reduced
spaces.
