  The so-called Mom-structures on hyperbolic cusped 3-manifolds without
boundary were introduced by Gabai, Meyerhoff, and Milley, and used by them to
identify the smallest closed hyperbolic manifold. In this work we extend the
notion of a Mom-structure to include the case of 3-manifolds with non-empty
boundary that does not have spherical components. We then describe a certain
relation between such generalized Mom-structures, called protoMom-structures,
internal on a fixed 3-manifold N, and ideal triangulations of N; in addition,
in the case of non-closed hyperbolic manifolds without annular cusps, we
describe how an internal geometric protoMom-structure can be constructed
starting from Epstein-Penner or Kojima decomposition. Finally, we exhibit a set
of combinatorial moves that relate any two internal protoMom-structures on a
fixed N to each other.
