On some questions and conjectures in the combinatorial theory of semigroups (Q1345736)

A semigroup \(S\) is repetitive if for each morphism \(\varphi : A^ + \to S\) and each integer \(k\), each long enough word contains \(k\) consecutive factors which are equal \(\text{mod }\varphi\). The authors show that there exists a (non abelian) group which is non repetitive and all whose proper subgroups are repetitive. They use a result of Ol'shanskij answering a question of Schmidt. They also show that a non repetitive semigroup, all whose subsemigroups are repetitive, must be a group equal to its derived subgroup. Moreover, a finitely generated semigroup, with linear growth and which is left cancellable, is repetitive. If \(\varphi : A^ + \to S\), \(A\) finite, is a repetitive morphism and if \(S\) satisfies a certain property \(\Sigma\), then \(S\) is repetitive. Many other results and problems are discussed.