Bulletin of the Australian Mathematical Society

Table of Contents - 1991 - Volume 44, Issue 03  

Research Article

The primes of S(R)

K.D. Magill Jra1

a1 Department of Mathematics, SUNY at Buffalo, 106 Diefendorf Hall, Buffalo NY 14214-3093, United States of America

Abstract

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 S0004972700029920_inline1 from R onto R such that m ≤ 1 + n/2 and

S0004972700029920_eqnU1

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)