Isomorphisms of lattices of subalgebras of semirings of continuous nonnegative functions with the max-plus

Isomorphisms $\varphi$ of semirings $C^\vee(X)$ of continuous nonnegative functions over an arbitrary Hewitt space $X$ with the condition $\varphi(\mathbb R^+)=\mathbb R^+$ are characterized in this work. It is proved that any isomorphism of lattices of all subalgebras of semirings $C^\vee(X)$ and $C^\vee(Y)$ is induced by a unique isomorphism of semirings excepting the case of one- and two-point Tychonovization of spaces.