Rank properties of certain semigroups. (Q1014785)

A subset \(U\) of a finite semigroup \(S\) is called independent if for every \(u\in U\) the element \(u\) does not belong to the semigroup \(\langle U\setminus\{u\}\rangle\) generated by the other elements of \(U\). For the semigroups \(CL_n\) (chain with \(n\) elements), \(CL_m\times CL_n\), \(Z_n\) (zero semigroup with \(n\) elements) and \(SL_n\) (free semilattice generated by \(n\) elements) the following ranks are found: \(r_1(S)=\max\{k:\forall U\subseteq S\), \(|U|=k\), \(U\) is independent\}, \(r_2(S)=\min\{k:\forall U\subseteq S\), \(|U|=k\), \(\langle U\rangle=S\}\), \(r_3(S)=\max\{k:\exists U\subseteq S\), \(|U|=k\), \(\langle U\rangle=S\), \(U\) is independent\}, \(r_4(S)=\max\{k:\exists U\subseteq S\), \(|U|=k\), \(U\) is independent\}, \(r_5(S)=\min\{k:\forall U\subseteq S\), \(|U|=k\), \(\langle U\rangle=S\}\).