Isomorphisms of semirings of continuous nonnegative functions with max-addition and isomorphisms of lattices of their subalgebras

Let $\mathbb{R}^{\vee}_+$ be the semifield with zero of nonnegative real numbers with operations of max-addition and multiplication and $C^{\vee}(X)$ be the semiring of continuous $\mathbb{R}^{\vee}_+$-valued functions on an arbitrary topological space $X$ with pointwise operation max-addition and multiplication. We call a subset $A\subseteq C^{\vee}(X)$ a subalgebra of the semiring $C^{\vee}(X)$ if $f\vee g,$ $fg,$ $rf\in A$ for any $f, g\in A$ and $r\in\mathbb{R}^{\vee}_+.$ For arbitrary topological spaces $X$ and $Y,$ we describe isomorphisms of the lattices of subalgebras (subalgebras with unity) of the semirings $C^{\vee}(X)$ and $C^{\vee}(Y).$