a1 Department of Mathematics, SUNY at Buffalo, 106 Diefendorf Hall, Buffalo NY 14214-3093, United States of America
S(R) is the semigroup, under composition, of all continuous selfmaps of the space R of real numbers. We show that the primes of S(R) are precisely those continuous selfmaps which are surjective and have exactly two local extrema. Additional results are then derived from this. For example, if f is any surjective continuous selfmap of R with n ≥ 2 local extrema, then there exist homeomorphisms from R onto R such that m ≤ 1 + n/2 and
where P is the polynomial defined by P(x) = x3 − x. It follows from this that the homeomorphisms together with the polynomial P generate a dense subsemigroup of S(R) where the topology on S(R) is the compact-open topology.
(Received December 03 1990)