In this question I have defined the following variety.
Let $(S, \cdot, e)$ be such that $(S, \cdot)$ is a semigroup, $e$ is a binary operation, and let the identities $e(x, y)x \approx x$, $e(x, y)\approx e(y, x)$ hold. Let's call a structure which satisfies those a double left monoid, or dlm.
One can see that if $(S, \cdot)$ is a left monoid with left identity $f$, then setting $e(x, y)\equiv f$ we get a dlm.
If $(S, \cdot, e)$, as a semigroup, isn't a left monoid, then it can't be a right monoid. Clearly, if $f$ were the right identity, then $e(x, f)f = f = e(x, f)$ for all $x$, and so $fx = x$ for all $x$, so it would be a monoid.
Is any dlm necessarily a left monoid after the transformation $(S, \cdot, e)\mapsto (S, \cdot)$ which forgets the operation $e$?
