  In a previous work, (dual)-$\mathfrak{m}$-Rickart lattices were studied. Now,
in this paper, we introduce $\mathfrak{m}$-endoregular lattices as those
lattices $\mathcal{L}$ such that $\mathfrak{m}$ is a regular monoid, where
$\mathfrak{m}$ is a submonoid with zero of End$_{lin}(\mathcal{L})$. We show
that these lattices can be characterized in terms of $\mathfrak{m}$-Rickart and
$\mathfrak{m}$-dual-Rickart lattices. Also, we compare these new lattices with
those lattices in which every compact element is a complement. We characterize
the $\mathfrak{m}$-endoregular lattices such that every idempotent in
$\mathfrak{m}$ is central in $\mathfrak{m}$ and we show that for these lattices
the complements are a sublattice which is a Boolean algebra. We introduce two
new concepts, $\mathfrak{m}$-$\mathcal{K}$-extending and
$\mathfrak{m}$-$\mathcal{T}$-lifting lattices. For these lattices, we show that
the monoid $\mathfrak{m}$ has a regular quotient monoid provided they satisfy
$\mathfrak{m}$-$C_2$ and $\mathfrak{m}$-$D_2$ respectively.
