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

Denote by R+∨\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb{R}_+^\vee$$\end{document} the semifield with zero of nonnegative real numbers with operations of max-addition and multiplication. Let X be a topological space and C∨(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∨(X) such that f ∨ g, fg, rf ∈ A for any f, g ∈ A, r∈R+∨\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r \in \mathbb{R}_+^\vee$$\end{document}. We consider the lattice A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb{A}$$\end{document}(C∨(X)) of subalgebras of the semiring C∨(X) and its sublattice A1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb{A}_1$$\end{document}(C∨(X)) of subalgebras with unity. The main result of the paper is the proof of the definability of the semiring C∨(X) both by the lattice A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb{A}$$\end{document}(C∨(X)) and by its sublattice A1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb{A}_1$$\end{document}(C∨(X)).