Largest subsemigroups of the full transformation monoid. (Q942114)

Let \(T_n\) denote the full transformation semigroup on an \(n\)-element set. In the paper under review the authors prove the following main results: (1) The largest size of a left zero subsemigroup of \(T_n\) is \(3^{(n/3)}\) if \(3\mid n\); \(4\cdot 3^{(n-4)/3}\) if \(3\mid n-1\); and \(2\cdot 3^{(n-2)/3}\) if \(3\mid n-2\). (2) The largest completely simple subsemigroup of \(T_n\) is the symmetric group \(S_n\). (3) The largest size of an inverse subsemigroup of \(T_n\) is \(\sum_{m=0}^{n-1}\binom{n-1}{m}^2m!\). Example of largest size subsemigroups are given.