Continuous right zero homomorphisms of \(U(S)\) (Q409512)

For a discrete semigroup \(S\), the Stone-Čech compactification \(\beta S\) has a natural structure of right topological semigroup, see [\textit{N. Hindman} and \textit{D. Strauss}, Algebra in the Stone-Čech compactification: theory and applications. de Gruyter Expositions in Mathematics. 27. Berlin: Walter de Gruyter (1998: Zbl 0918.22001)]. If \(S\) is cancellative then the remainder \(S^* = \beta S \setminus S\) is a subsemigroup of \(\beta S\).NEWLINENEWLINEBy the definition, a mapping \(\pi: T \to X\) of a semigroup \(T\) into a set \(X\) is a right zero homomorphism if \(\pi(pq)=\pi(q)\) for all \(p,q\in T\).NEWLINENEWLINEUsing slowly oscillating functions, the authors prove the following elegant statement.NEWLINENEWLINELet \(S\) be a countable cancellative semigroup, and \(X\) a connected compact metric space. Then there exists a continuous surjective right zero homomorphism \(\pi : S^* \to X\).NEWLINENEWLINEReviewer's remark. Lemma 2.3 and Lemma 2.4 are partial cases of Lemma 6.47 from [loc. cit.].