  We introduce the concept of isolated factorizations of an element of a
commutative monoid and study its properties. We give several bounds for the
number of isolated factorizations of simplicial affine semigroups and numerical
semigroups. We also generalize $\alpha$-rectangular numerical semigroups to the
context of simplicial affine semigroups and study their isolated
factorizations. As a consequence of our results, we characterize those complete
intersection simplicial affine semigroups with only one Betti minimal element
in several ways. Moreover, we define Betti sorted and Betti divisible
simplicial affine semigroups and characterize them in terms of gluings and
their minimal presentations. Finally, we determine all the Betti divisible
numerical semigroups, which turn out to be those numerical semigroups that are
free for any arrangement of their minimal generators.
