Abstract
Let \(S\) be the semifield \(\mathbb{R}_{+}\) with zero of nonnegative real numbers or the semifield \(\mathbb{P}\) of positive real numbers. The set of all continuous functions \(f\colon X\to S\) with pointwise operations of addition and multiplication of functions defined on an arbitrary topological space \(X\) forms the semiring \(C(X,S).\) By a subalgebra we mean a nonempty subset \(A\) of \(C(X,S)\) such that \(f+g,fg,rf\in A\) for any \(f,g\in A\) and any \(r\in S.\) For arbitrary topological spaces \(X\) and \(Y,\) we describe isomorphisms of the semirings \(C(X,S)\) and \(C(Y,S)\) and isomorphisms of the lattices of their subalgebras (subalgebras with unity).
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The paper was prepared within the framework of the state comission of the Ministry of Science and Higher Education of the Russian Federation, project no. 1.5879.2017/8.9.
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(Submitted by M. M. Arslanov)
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Sidorov, V.V. Isomorphisms of Semirings of Continuous Nonnegative Functions and the Lattices of Their Subalgebras. Lobachevskii J Math 41, 1684–1692 (2020). https://doi.org/10.1134/S1995080220090255
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DOI: https://doi.org/10.1134/S1995080220090255