Suppose that $a$ belongs to the finite semigroup $S$ (especially semigroups of transformations). Are there any techniques for determining the cardinality of $SaS$?
- Example: Let $S=\mathcal{T}_n$ (full transformation semigroup). We can write each idempotent $e= \left( \begin{array}{ccc} A_1 & A_2 & \cdots & A_r \\ a_1 & a_2 & \cdots & a_r \end{array} \right)(a_i\in A_i)$.
I'm wondering if $\mathcal{T}_ne\mathcal{T}_n$ is a helpful structure for computing the cardinality $\mathcal{T}_ne\mathcal{T}_n$ for each idempotent $e$. I'm not interested in the answer to this example.
Your question is not very precise since you don't say how $S$ is given: is it the semigroup generated by a given set of transformations, is the multiplication table already available, etc?
In any case, I invite you to read carefully the following article, which contains a number of useful algorithms and an extended bibliography on the topic:
J. East, A. Egri-Nagy, J.D. Mitchell, Y. Péresse, Computing finite semigroups, J. Symbolic Comput. 92 (2019), 110--155.
Note that if $S$ is a monoid, $SaS$ is the ideal generated by $a$. If $X$ is a generating set for $S$, you can compute it by constructing a sort of double-sided Cayley graph with root $a$ and edges of the form $s \xrightarrow{x} sx$ and $s \xrightarrow{x} xs$, where $s \in S$ and $x \in X$.