Automorphisms of the semigroup End(K[x 1 , . . . , x n ])

What is it about?

The authors describe the automorphism group of the endomorphism monoid ${\rm End}(K[x_1,\dots,x_n])$, where $K$ is a field. The main result is that every such automorphism $\Phi$ is semi-inner. An automorphism $\Phi$ is called quasi-inner if there exists an adjoined bijection $s$ of ${\rm End}(K[x_1,\dots,x_n])$ such that $\Phi(\nu)=s\nu s^{-1}$ for any $\nu\in {\rm End}(K[x_1,\dots,x_n])$. A quasi-inner $\Phi$ is called semi-inner if there exists a field automorphism $\delta$ of $K$ such that $(\delta,s)$ is a semi-linear automorphism. The result is transferred to the category of finitely generated, free, commutative and associative algebras of the variety of commutative algebras. Reviewed by Ulrich Knauer

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