On homomorphisms from semigroups onto cyclic groups (Q1300593)

Let \(S\) be a semigroup, \(T\) be a subset of \(S\) and \(n\) be a positive integer. Define: \(\widehat T=\{x\in S\mid TxT\cap T\neq\emptyset\}\), \(T_l=\{x\in S\mid Tx\cap T\neq\emptyset\}\), \(T_r=\{x\in S\mid xT\cap T\neq\emptyset\}\), \(\overline T=T_l\cap T_r\). Let \(S_n\) be the set of all elements \(x\in S\) for which there exist elements \(s_1,\dots,s_k\) such that \(x\) is a product of \(n\) copies of \(s_1\), \(n\) copies of \(s_2\), \dots, \(n\) copies of \(s_k\) not necessarily in this order. In this paper a characterization of semigroups that have a cyclic group as homomorphic image is given. It is proved: Theorem. For every semigroup \(S\) and every integer \(m\geq 2\) the following are true: 1) There exists a nontrivial homomorphism \(f\colon S\to\mathbb{Z}_m\) if and only if \(\overline S_m\neq S\). 2) There exists a surjective homomorphism \(f\colon S\to\mathbb{Z}_m\) if and only if \(\overline S_m\neq\overline S_{m/p}\) for every prime number \(p\) dividing \(m\).