A minimal variety is a variety whose only proper subvarieties are trivial. They form the atoms of subvariety lattices and as such are pretty important in UA. A key example of a minimal variety is the variety of semilattices.
But there is a compelling sense in which a minimal variety is not actually minimal.
Let’s define the notion of a plusquam-subvariety: the collection of algebra is an plusquam-subvariety of the variety if it consists exactly of the -reducts of some expansion of . Clearly, there exist proper nontrivial extra-subvarieties of minimal varieties: one may simply look at the semilattice reducts of distributive lattices, or even of Boolean algebra…
This brings up an obvious question: do there exist plusquam-minimal varieties, varieties with no proper nontrivial plusquam-subvarieties?
The intuitive answer is “uh, yeah, no”, considering how easy it is to construct plusquam-subvarieties, but it is not outside of the realm of my imagination that some pathological example can be constructed. Perhaps further knowledge will come with further delving into the depths of UA…