Let $S$ be a finite semi-group which basically means that it is an associative binary operation.
$A$ will be called as a generating set of $S$ if every element can be written as a multiplication of elements of $A$.
If $S$ is a group then there exists a $A$ of size $\log |S|$. I know previous statement is not true for semi-group $S$.
What is the size of smallest $A$(in terms of number of elements in it) which generates semi-group $S$? Note that $S$ is not a zero semi-group.
Consider the set $S_n= \{e_1,...,e_n\}\cup \{x_0\}$ of $n+1$-elements. Define $$e_i\cdot e_j = \begin{cases}x_0&\mbox{ if }i\neq j\\ e_i &\mbox{ if }i=j \end{cases}$$ for any $1\leq i,j\leq n$ and $$x_0 \cdot e_i = x_0 = e_i\cdot x_0$$ for every $1\leq i\leq n$. Then $\cdot$ is associative and moreover, the smallest set of generators $A_n$ for $S_n$ is $\{e_1,...,e_n\}$. Clearly $$\lim_{n\rightarrow +\infty}\frac{|S_n|}{|A_n|} = 1$$ This shows that there are no interesting results in this direction at this level of generality.
In the semigroup $S_n = \{x_1, \ldots, x_n\}$ with the product defined by $x_ix_j = x_j$ for all $i$ and $j$, the smallest set of generators is $S_n$ itself.
