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

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## 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*)).

## Keywords and phrases

semiring of continuous functions subalgebra lattice of subalgebras isomorphism Hewitt space max-addition## Preview

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