Let $P$ be a poset with a monoidal structure respecting the poset structure. This means there is an operation $P \times P \to P$ such that $a \leq b \implies ac \leq bc$.
As usual, call a poset complete if $P$ admits limits and colimits (in the sense of category theory). This means that one can always find an inf and a sup to any subset of $P$. (In other words, $P$ is a complete lattice.) Call $P$ dense if for any two $a <b$, there exist $c$ so that $ a <c< b$.
What is the initial dense, complete, monoidal poset containing a copy of the integers? Must it be the real numbers adjoin $\pm \infty$?
More generally, is there an adjunction between the category of monoidal posets, and the category of monoidal dense complete posets?