Is the set of identities of a single binary operation densely ordered under this strict partial order?

Question

This is somewhat similar to my previous question, here: Is the partial order of equational theories of a single binary operation dense?, but slightly different. Consider the set $S$ of identities of a single binary operation $\{*\}$. So, for example, the commutative identity $(x*y)=(y*x)$ is in $S$. I define a relation $<$ on $S$ by saying that, given identities $E$ and $E'$ in $S$, $E < E'$ holds iff $E$ can be derived from $E'$, but not vice versa. It is easy to prove that $<$ is a strict partial order. My question is, is the order dense? That is, if $E < E'$, does there exist a $E''$ in $S$ such that $E<E''<E'$?

Answer 1

The answer is No, the order is not dense.

Let $E'$ be the identity $x=y$ and let $E$ be the identity $$ (x*((y*x)*x))*(y*(z*x))=y. $$ Then $E<E'$ and there is no strictly intermediate identity $E''$. The reason for this is that $E'$ axiomatizes the trivial variety in this language (as is easily seen) while $E$ axiomatizes a minimal variety in this language (which is not easily seen).

For the not-easily-seen part, I am using one of the results of the paper

Short Single Axioms for Boolean Algebra
William McCune, Robert Veroff, Branden Fitelson, Kenneth Harris, Andrew Feist, Larry Wos
Journal of Automated Reasoning volume 29, pages 1-16 (2002)

They prove that $E$ is the shortest single axiom for the Sheffer stroke (NAND or NOR) operation.