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.
Categories: math.CT math.RA