Working on semi orders, for a binary relation $R$ on a set $A$ we have that it is a semi order if the following holds
$$ aRb\longleftrightarrow u(a)\geq u(b) + q $$
For $q\geq 0$ and $u : A \to\mathbb{R}$.
Then it is written in my book that
$\neg(aRb)$ is equivalent to
$$ u(a)<u(b) - q $$
Whereas I was thinking of just $u(a)<u(b)+q$.
Have you some explanations to provide on this please ?
Your thinking is correct. The negation of a statement $x \geq y$ for any $x,y\in\bf R$ is $x<y$.
Now the dual, $a R^d b$, of the statement is another thing. It’s defined so that $a R^d b$ precisely when $b R a$. Thus, for example, $\subseteq$ and $\supseteq$ are dual to one another. And for your example where $aRb$ is defined in terms of $u$ and $q$ to mean $u(a)\geq u(b)+q$, the dual is of course $u(a) \leq u(b)-q$. But even that is not (quite) what your book said the negation is.