Automorphisms of the lattice of all subalgebras of the semiring of polynomials in one variable

Abstract

In this paper, we describe automorphisms of the lattice ${\mathbb A}$ of all subalgebras of the semiring ℝ⁺[x] of polynomials in one variable over the semifield ℝ⁺ of nonnegative real numbers. It is proved that any automorphism of the lattice A is generated by an automorphism of the semiring ℝ⁺[x] that is induced by a substitution x ⟼ px for some positive real number p. It follows that the automorphism group of the lattice ${\mathbb A}$ is isomorphic to the group of all positive real numbers with multiplication. A technique of unigenerated subalgebras is applied.