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

Abstract

Denote by \(\mathbb{R}_+^\vee\) 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, \(r \in \mathbb{R}_+^\vee\). We consider the lattice \(\mathbb{A}\)(C^∨(X)) of subalgebras of the semiring C^∨(X) and its sublattice \(\mathbb{A}_1\)(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 \(\mathbb{A}\)(C^∨(X)) and by its sublattice \(\mathbb{A}_1\)(C^∨(X)).