On the endomorphism monoids of (uniquely) complemented lattices

Let $L$ be a lattice with $0$ and $1$. An endomorphism $\varphi$ of $L$ is a $\{0,1\}$-endomorphism, if it satisfies $0\varphi = 0$ and $1\varphi = 1$. The $\{0,1\}$-endomorphisms of $L$ form a monoid. In 1970, the authors proved that every monoid $\mathcal M$ can be represented as the $\{0,1\}$-endomorphism monoid of a suitable lattice $L$ with $0$ and $1$. In this paper, we prove the stronger result that the lattice $L$ with a given $\{0,1\}$-endomorphism monoid $\mathcal M$ can be constructed as a uniquely complemented lattice; moreover, if $\mathcal M$ is finite, then $L$ can be chosen as a finite complemented lattice.