Automorphisms of the semiring of polynomials $\mathbb{R}_+^{\vee}[x]$ and lattices of its subalgebras

V. V. Sidorov

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 26 (2020) no. 3, pp. 171-186

Résumé

A commutative semiring with zero and unity different from a ring where each nonzero element is invertible is called a semifield with zero. Let $\mathbb{R}^{\vee}_+$ be the semifield with zero of nonnegative real numbers with operations of max-addition and multiplication. For any positive real numbers $a$ and $s$, denote by $\psi_{a,s}$ the automorphism of the semiring of polynomials $\mathbb{R}_+^{\vee}[x]$ defined by the rule $\psi_{a, s}\colon a_0\vee a_1x\vee\ldots\vee a_nx^n\mapsto a_0^s\vee a_1^s(ax)\vee\ldots\vee a_n^s(ax)^n$. It is proved that the automorphisms of the semiring $\mathbb{R}_+^{\vee}[x]$ are exactly the automorphisms $\psi_{a, s}$. The ring $C(X)$ of continuous $\mathbb{R}$-valued functions defined on an arbitrary topological space $X$ is an algebra over the field $\mathbb{R}$ of real numbers. A subalgebra of $C(X)$ is any nonempty subset closed under addition and multiplication of functions and under multiplication by constants from $\mathbb{R}$. Similarly, we call a nonempty subset $A\subseteq \mathbb{R}_+^{\vee}[x]$ a subalgebra of $\mathbb{R}_+^{\vee}[x]$ if $f\vee g,fg,rf\in A$ for any $f, g\in A$ and $r\in\mathbb{R}^{\vee}_+$. It is proved that an arbitrary automorphism of the lattice of subalgebras of $\mathbb{R}_+^{\vee}[x]$ is induced by some automorphism of $\mathbb{R}_+^{\vee}[x]$. The same result also holds for the lattice of subalgebras with unity of the semiring $\mathbb{R}_+^{\vee}[x]$. The technique of one-generated subalgebras is applied.

Détail

Keywords: semiring of polynomials, lattice of subalgebras, max-addition.
Mots-clés : automorphism