Is the partial order of equational theories of a single binary operation dense?

Question

Let our signature be a single binary operation $\{*\}$. Consider the set of equational theories of that signature, partially ordered by inclusion. Is that partial order dense?

Answer 1

The set of equational theories in a fixed language under the inclusion order forms an algebraic lattice. Any algebraic lattice of more than one element will have elements $<$ with no element strictly between $$ and $$. (Choose $$ compact and $$ a lower cover of $$.)