Universality of Small Lattice Varieties
Vácłav Koubek, Jiří Sichler · Proceedings of the American Mathematical Society · 1984
There exists a finitely generated lattice variety $S$ such that the class of all nonconstant homomorphisms between members of $S$ contains a universal category as a full subcategory. In particular, every monoid $M$ is isomorphic to the monoid of all nonconstant endomorphisms of a lattice from $S$, and $S$ contains arbitrarily large lattices representing $M$. The category of all $(0,1)$-homomorphisms of lattices in $S$ is also shown to be universal.