According to Rosales and Garcia, an integer is a pseudo-frobenius number if $x \not \in S$ and $x + s \in S$ for all $s \in S \setminus \{0 \}$ without the 0 element. Furthermore, let $a \leq_S b$ if $b - a \in S$. The book claims that the set of Pseudo-frobenius numbers are the maximal elements with respect to $\leq_S$ of $\mathbb{Z} \setminus S$.
I am confused what it means for the pseudo-frobenius numbers to be "the maximal elements with respect to $\leq_S$ of $\mathbb{Z} \setminus S$." Initially, I thought this statement claimed that the Pseudo-frobenius numbers are the largest elements of $\mathbb{Z} \setminus S$ such that for any $b \in S$ and $a \in \mathbb{Z} \setminus S$, we have that $a \leq_S b$. However, this doesn't seem to be necessarily true. Could someone please clarify?
In the context of partial orders, largest is not the same as maximal: maximal means that there is not bigger, while largest is bigger than everything.
An element $x$ is maximal with respect to the partial order $\le_S$ on a set $U:=\Bbb Z\setminus S$, if $x\in U$ and for all $y\in U,\ x\le_S y$ implies $x=y$.
Now $x\in U$ means that $x\notin S$, and for any $s\in S\setminus\{0\}$, the element $y=x+s$ can't be in $U$ (as then it would $x\ne y,\ x\le_S y$), so $$\forall s\in S: x+s\in S\,.$$