Let
(
L
,
M
)
(\mathfrak {L},M)
be a local Noether lattice. If the maximal element
M
M
is meet principal, it is well known and easily seen that every element of
L
\mathfrak {L}
is meet principal. In this note, we obtain the corresponding result for
M
M
join-principal. We also consider join-principal elements generally under the assumption of the weak union condition and show, for example, that the square of a join-principal element is principal.