Definability of Hewitt spaces by the lattices of subalgebras of semifields of continuous positive functions with max-plus

The lattice $\mathbb{A}(U^{\vee}(X))$ of subalgebras of the semifield $U^{\vee}(X)$ of all continuous positive functions defined on a topological space $X$ is considered. A topological space is said to be a Hewitt space if it is homeomorphic to a closed subspace of a Tychonoff power of the real line $\mathbb{R}$. The main result of the paper is the proof of the fact that any Hewitt space $X$ is determined by the lattice $\mathbb{A}(U^{\vee}(X))$.