The determinability of compacts by lattices of ideals and congruencies of semirings of continuous $[0,1]$-valued functions on them

We consider an idempotent semiring of continuous $[0,1]$-valued functions defined on a compact $X$ with the usual multiplication and addition $\max$. We prove the determinability of $X$ by the lattice of ideals and the lattice of congruencies of the indicated semiring.