Determinability of Semirings of Continuous Nonnegative Functions with Max-Plus by the Lattices of Their Subalgebras

Denote by R-+(V) the semifield with zero of nonnegative real numbers with operations of max-addition and multiplication. Let X be a topological space and C-V(X) be the semiring of continuous nonnegative functions on X with pointwise operation max-addition and multiplication of functions. By a subalgebra we mean a nonempty subset A of C-V(X) such that f V g, fg, rf is an element of A for any f, g is an element of A, r is an element of R-+(V). We consider the lattice A(C-V(X)) of subalgebras of the semiring C-V(X) and its sublattice A(1)(C-V(X)) of subalgebras with unity. The main result of the paper is the proof of the definability of the semiring C-V(X) both by the lattice A(C-V(X)) and by its sublattice A(1)(C-V(X)).