For any semigroup $S$, let $A$ be a non-empty subset of $S$. Then the subsemigroup generated by $A$ that is, the smallest subsemigroup of $S$ containing $A$, is denoted by $\langle A\rangle$. If there exists a finite subset $A$ of $S$ such that $S=\langle A\rangle$, then $S$ is called a finitely generated semigroup. The rank of a finitely generated semigroup $S$ is defined by $${\rm rank}(S)=\min \{\, |A|:\langle A\rangle =S\}.$$
Let $T$ be a subsemigroup of any finite semigroup $S$ and $S\setminus T$ be subsemigroup of $S$. If we know the rank of $T$, then can we determine the rank of $S\setminus T$ ?
No way. Take any finite semigroup $S$ and a new element $0$. Form a new semigroup $S^0$ with support $S \cup \{ 0\}$ and product defined by $s0 = 0s = 0$ for all $s \in S^0$. In other words, you add a new zero to $S$. Then $T = \{0\}$ is a subsemigroup of $S^0$ and its rank is $1$. Furthermore, $S = S^0 \setminus T$ is a subsemigroup of $S^0$ and its rank can be anything, depending on the rank of $S$.