  In this paper, we study the atomic structure of Puiseux monoids generated by
monotone sequences. To understand this atomic structure, it is often useful to
know whether the monoid has a bounded generating set. We provide necessary and
sufficient conditions for the atomicity and boundedness to be transferred from
a monotone Puiseux monoid to all its submonoids. Finally, we present two
special subfamilies of monotone Puiseux monoids and fully classify their atomic
structure.
