Isomorphisms of lattices of subalgebras of semirings of continuous nonnegative functions

In this work, lattice isomorphisms of semirings $C^+(X)$ of continuous nonnegative functions over an arbitrary topological space $X$ are characterized. It is proved that any isomorphism of lattices of all subalgebras with a unit of semirings $C^+(X)$ and $C^+(Y)$ is induced by a unique isomorphism of semirings. The same result is also correct for lattices of all subalgebras excepting the case of two-point Tychonovization of spaces.