Some monothetic subsemigroups of \(S(\mathbb{R})\)
Let \(R^n\) denote the \(n\)-dimensional Euclidean topological space and denote by \(S(R^n)\) the semigroup, under composition, of all continuous selfmaps of \(R^n\). It has been known for a long time that \(S(R^n)\) contains a copy of every finite semigroup if \(n>1\). This is not at all the case for the semigroup \(S(R)\), of all continuous selfmaps of a space \(R\) of real numbers. For example, the only finite subgroups of \(S(R)\) are the cyclic groups of orders one and two. In general, however, not much seemed to have been known about the finite subsemigroups of \(S(R)\). For \(f\in S(R)\), denote by \(C(f)\) the cyclic subsemigroup which is generated by \(f\). The reviewer posed [in `Some properties of \(S(R)\)', Proc. Monash Conf. on Semigroup Theory (in honor of G. B. Preston) Ed. by T. E. Hall, P. R. Jones and J. C. Meakin, World Scientific, Inc., New Jersey, 163-179 (1991)]\ the problem of characterizing those functions \(f\in S(R)\) for which \(C(f)\) is a finite semigroup and conjectured that for a certain ``steplike'' function \(f_m\) with \(m\) ``steps'' where \(m\) is a positive integer, the order of the semigroup \(C(f_m)\) is \(2r+5\) where \(2^r\leq m< 2^{r+1}\) and \(r\) is a positive integer. In the main result of the paper, the author shows that this conjecture is true. He also determines that the order of \(C(f_m\circ f_m)\) is \(r+2\) which means that \(S(R)\) contains a subsemigroup of order \(n\) for each positive integer \(n\), a fact which heretofore did not seem to be known.
scientific article; zbMATH DE number 836813