Definability of semifields of continuous positive functions by the lattices of their subalgebras

We consider the lattice $\mathbb{A}(U(X))$ of subalgebras of a semifield $U(X)$ of continuous positive functions on an arbitrary topological space $X$ and its sublattice $\mathbb{A}_1(U(X))$ of subalgebras with unity. The main result of the paper is the proof of the definability of any semifield $U(X)$ both by the lattice $\mathbb{A}(U(X))$ and by its sublattice $\mathbb{A}_1(U(X))$.
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