By a Dedekind cut, we mean, an ordered pair $(L,U)$ of subsets of $\mathbb{Q}$ such that they are disjoint, their union is $\mathbb{Q}$, and
Each member of $L$ is smaller than each member of $U$
$L$ contains no largest rational number.
Let $(L,U)$ be a Dedekind cut for which $L$ contains some positive rationals.
Let $L'$ be the collection of non-positive rationals, along with those positive rationals $x$ whose product with all positive rationals of $L$ is $<1$. Let $U'$ be the complement of $L'$ in $\mathbb{Q}$.
Can we say that $(L',U')$ defined in this way is the multiplicative inverse of $(L,U)$?
(One may see this wiki-link for product of Dedekind cuts).