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

In this paper, we describe automorphisms of the lattice $\mathbb A$ of all subalgebras of the semiring $\mathbb R^+[x]$ of polynomials in one variable over the semifield $\mathbb R^+$ of nonnegative real numbers. It is proved that any automorphism of the lattice $\mathbb A$ is generated by an automorphism of the semiring $\mathbb R^+[x]$ that is induced by a substitution $x\mapsto px$ 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.