Isomorphisms of lattices of subalgebras of the semifield of continuous positive functions with max-addition

Let $\mathbb{P}^{\vee}$ be the semifield of positive real numbers with operations of max-addition and multiplication and $U^{\vee}(X)$ be the semifield of continuous $\mathbb{P}^{\vee}$-valued functions on an arbitrary topological space $X$ with pointwise operation max-addition and multiplication. We call a subset $A\subseteq U^{\vee}(X)$ a subalgebra if $f\vee g,$ $fg,$ $rf\in A$ for any $f, g\in A,$ $r\in\mathbb{P}^{\vee}.$ We describe isomorphisms of lattices of subalgebras of semifields $U^{\vee}(X).$