Are there two different binary operations on a set with at least two elements that do not enmesh?

Question

Consider an algebraic structure $(S,+,*)$, where $+$ and $*$ are simply two different binary operations, not necessarily addition or multiplication. We say that $+$ and $*$ do not enmesh if the equational identities of $(S,+)$ united with the equational identities of $(S,*)$ suffice to generate all the equational identities of $(S,+,*)$. We say they do enmesh if it is not the case that they do not enmesh. For example, over the reals, $+$ and $*$ do enmesh, because of the distributive law connecting the two. I am finding it hard to come up with two explicit binary operations that do not enmesh. Can someone give an example, preferably two operations on a finite set, preferably as small a finite set as possible?

Answer 1

I can think of an example which might be considered too trivial, in a way...

The variety of Left-zero semigroups (given by the identity $xy=x$) is such that the free algebra on $n$ elements has $n$ elements. The same applies to Right-zero semigroups.
So take an $n$-element set ($n=2,3,\ldots$ whatever) and make $x+y=x$ and $x*y=y$.
The free algebra on that set has the same number of elements (it's trivial to check that it has the universal mapping property).

So an equational base for the resulting variety is given by those two identities.