  Let $L$ be a full-rank lattice in $\mathbb R^d$ and write $L^+$ for the
semigroup of all vectors with nonnegative coordinates in $L$. We call a basis
$X$ for $L$ positive if it is contained in $L^+$. There are infinitely many
such bases, and each of them spans a conical semigroup $S(X)$ consisting of all
nonnegative integer linear combinations of the vectors of $X$. Such $S(X)$ is a
sub-semigroup of $L^+$, and we investigate the distribution of the gaps of
$S(X)$ in $L^+$, i.e. the points in $L^+ \setminus S(X)$. We describe some
basic properties and counting estimates for these gaps. Our main focus is on
the restrictive successive minima of $L^+$ and of $L^+ \setminus S(X)$, for
which we produce bounds in the spirit of Minkowski's successive minima theorem
and its recent generalizations. We apply these results to obtain analogous
bounds for the successive minima with respect to Weil height of totally
positive sub-semigroups of ideals in totally real number fields.
