Minimal variety

Remarks

Example: the variety of Boolean algebras is minimal. Every nontrivial Boolean algebra contains the two-element Boolean algebra $\mathbf{2}$ as a subalgebra, and $\mathbf{2}$ generates the variety of all Boolean algebras. So a subvariety with a nontrivial member contains $\mathbf{2}$ and hence every Boolean algebra. Non-example: the variety of all groups is not minimal. Its subvariety of abelian groups contains $\Z/2\Z$, so it is not trivial, and it is a proper subvariety. In the lattice of subvarieties of a variety, the minimal varieties are exactly the atoms, the elements that cover the trivial variety.