Multiplicative bijections of semigroups of interval-valued continuous functions

TL;DR

This paper characterizes certain topological spaces based on the structure of multiplicative bijections on semigroups of interval-valued continuous functions, providing a counterexample to a previous conjecture.

Contribution

It offers a complete characterization of spaces where multiplicative bijections have a specific form, disproving a conjecture by Marovt.

Findings

Characterization of spaces with specific multiplicative bijections

Disproof of Marovt's conjecture

Extension to other semigroups of functions

Abstract

We characterize all compact and Hausdorff spaces $X$ which satisfy that for every multiplicative bijection $\phi$ on $C(X, I)$, there exist a homeomorphism $\mu : X \to X$ and a continuous map $p: X \to (0, +\infty)$ such that $$\phi (f) (x) = f(\mu (x))^{p(x)}$$ for every $f \in C(X,I)$ and $x \in X$. This allows us to disprove a conjecture of Marovt (Proc. Amer. Math. Soc. {\bf 134} (2006), 1065-1075). Some related results on other semigroups of functions are also given.