A unital quantale is a suplattice with a compatible monoid structure. A quantale is called idempotent if it is idempotent as a monoid (every element is idempotent) (analogously for commutativity). Every suplattice can be regarded as a unital, commutative, idempotent suplattice, by chosing the join as operation. In this quantale the bottom element $\bot$ is the identity element. Does the converse hold, i. e.: Given a commutative, idempotent unital quantale (with an operation $+$) where $\bot$ is the identity, is $+$ the same as the join $\vee$?
According to the nLab it is known that $+$ is the meet if you replace $\bot$ by $\top$ in the condition (does anybody know a reference for this statement?).
As noticed below, this question is actually about partially order monoids, not quantales (I was confused). I changed the title such that it can be found more easily. The answer and the comment imply that the described situation is actually the situation of the trivial lattice consisting of a single element. But it generalizes to partially ordered monoids.