Automorphisms of the endomorphism semigroup of a free commutative algebra

TL;DR

This paper characterizes the automorphism groups of endomorphism semigroups and categories of free commutative algebras over any field, solving two open problems and extending previous results.

Contribution

It provides a complete description of automorphisms of endomorphism semigroups and categories of free commutative algebras, generalizing known results to arbitrary fields.

Findings

Automorphism group of $ ext{End}(K[x_1,...,x_n])$ is described by semi-linear automorphisms.

The result applies to any field $K$, not just infinite fields.

Solves two open problems posed by B. Plotkin.

Abstract

We describe the automorphism group of the endomorphism semigroup $\End(K[x_1,...,x_n])$ of ring $K[x_1,...,x_n]$ of polynomials over an {\it arbitrary} field $K$. A similar result is obtained for automorphism group of the category of finitely generated free commutative-associative algebras of the variety $\mathcal{CA}$ commutative algebras. This solves two problems posed by B. Plotkin (\cite{24}, Problems 12 and 15). More precisely, we prove that if $\varphi\in \Aut\End(K[x_1,...,x_n])$ then there exists a semi-linear automorphism $s:K[x_1,...,x_n]\to K[x_1,...,x_n]$ such that $\varphi(g)=s\circ g\circ s^{-1}$ for any $g\in\End(K[x_1,...,x_n])$. This extends the result by A. Berzins obtained for an infinite field $K$.