Multiplicative bijections of semigroups of interval-valued continuous functions

Abstract

Abstract. We characterize all compact and Hausdorff spaces X which satisfy the condition that for every multiplicative bijection ϕ on C(X, I), there exist a homeomorphism µ : X −→ X and a continuous map p :for every f ∈ C(X, I) and x ∈ X. This allows us to disprove a conjecture of Marovt (Proc. Amer. Math. Soc. 134 (2006Soc. 134 ( ), 1065Soc. 134 ( -1075. Some related results on other semigroups of functions are also given.

Lattice isomorphisms between certain sublattices of continuous functions

“…The conjecture for multiplicative (and consequently for order preserving) bijections on C(X, I) was disproved by Ercan and Önal in [4]. The more general problem of characterizing those compact Hausdorff spaces X for which any multiplicative bijection T : C(X, I) −→ C(X, I) has the above standard form has been investigated in [1].…”

“…For a compact Hausdorff space X, we say that a subset A of C(X, I) has Urysohn's property, if for any pair of disjoint closed subsets F and G of X there exists f ∈ A such that f = 0 on F and f = 1 on G (compare with the Property 1 in [1]). We also say that the evaluation of A on X is dense in (respectively, equal to) I if for each x ∈ X, the set A x = {f (x) : f ∈ A} is dense in (respectively, equal to) I.…”