A [*unital quantale*](http://ncatlab.org/nlab/show/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?).


### Edit


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.



Commutative, idempotent partially ordered monoids
The answer is yes. Here is a proof. 


**Lemma 1**. Let $\leq$ be the suplattice order. 
If $a\leq b$ the $c+a\leq c+b$. 


Proof. $c+b=c+(a\vee b)=(c+a)\vee (c+b)$. 


By the standard equivalence betwee join semilattices and idempotent commutative semigroups it suffices to prove:


**Lemma 2.** $a\leq b\iff a+b=b$. 


Proof.
If $a\leq b$ then $b=b+\bot\leq b+a\leq b+b=b$ using Lemma 1. So $b=a+b$. 


If $a+b=b$ then $a=a+\bot\leq a+b=b$, again using Lemma 1. This completes the proof. 


