  A finite semigroup is finitely related (has finite degree) if its term
functions are determined by a finite set of finitary relations. For example, it
is known that all nilpotent semigroups are finitely related. A nilpotent monoid
is a nilpotent semigroup with adjoined identity. We show that every
$4$-nilpotent monoid is finitely related. We also give an example of a
$5$-nilpotent monoid that is not finitely related. To our knowledge, this is
the first example of a finitely related semigroup where adjoining an identity
yields a semigroup which is not finitely related. We also provide examples of
finitely related semigroups which have subsemigroups, homomorphic images, and
in particular Rees quotients, that are not finitely related.
