Isomorphisms of lattices of subalgebras of semifields of positive continuous functions

We consider the lattice of subalgebras of a semifield $U(X)$ of positive continuous functions on an arbitrary topological space $X$ and its sublattice of subalgebras with unity. We prove that each isomorphism of the lattices of subalgebras with unity of semifields $U(X)$ and $U(Y)$ is induced by a unique isomorphism of the semifields. The same result holds for lattices of all subalgebras excluding the case of the double-point Tychonoff extension of spaces.