Divisibility of certain periodic semigroups. (Q815181)

A semigroup \(S\) is called divisible if for any element \(x\) of \(S\) and for any positive integer \(n\) there is an element \(y\) of \(S\) such that \(x=y^n\). One of the main theorems: Assume \(S\) is periodic and the orders of elements are bounded. Then \(S\) is divisible if and only if it is a band. More emphasis is concerning some transformation semigroups. Let \(X\) be a non-empty set, and \(T_X\) the full transformation semigroup on \(X\). For \(\alpha\in T_X\) let \(\text{ran\,}\alpha\) denote the range of \(\alpha\). Let \(F(X)=\{\alpha\in T_X\mid|\text{ran\,}\alpha|<\infty\}\). It is known that \(F(X)\) is periodic. Define \(F_n(X)=\{\alpha\in T_X\mid|\text{ran\,}\alpha|\leq n\}\). Theorem. For any nonempty set \(X\), (1) \(F(X)\) is divisible if and only if \(|X|=1\). (2) \(F_n(X)\) is a periodic semigroup, and the orders of elements are bounded by \(n!\). If \(X\) is infinite then \(F_n(X)\) is infinite. Theorem. For any nonempty set \(X\) and for any positive integer \(n\), \(F_n(X)\) is divisible if and only if \(|X|=1\) or \(n=1\). The reviewer is stimulated by this paper, since the reviewer started divisible semigroups 40 years ago.