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Automorphisms of the endomorphisms semigroup of a free associative algebra. (English) Zbl 1152.08004

Summary: Let \(\mathcal A\) be the variety of associative algebras over a field \(K\) and \(A = K\langle x_1,\dots,x_n\rangle\) be a free associative algebra in the variety \(\mathcal A\) freely generated by a set \(X = \{x_1,\dots,x_n\}\), \(\text{End}\,A\) the semigroup of endomorphisms of \(A\), and \(\text{Aut\, End}\,A\) the group of automorphisms of the semigroup \(\text{End}\, A\). We prove that the group \(\text{Aut\,End\,} A\) is generated by semi-inner and mirror automorphisms of \(\text{End\,} A\). A similar result is obtained for the automorphism group \(\text{Aut\,}{\mathcal A}^\circ\), where \({\mathcal A}^\circ\) is the subcategory of finitely generated free algebras of the variety \(\mathcal A\). The latter result solves Problem 3.9 formulated by G. Mashevitzky, B. Plotkin and E. Plotkin [Electron. Res. Announc. Am. Math. Soc. 2002, 1–10 (2002; Zbl 0996.08008)].

MSC:

08A35 Automorphisms and endomorphisms of algebraic structures
08C05 Categories of algebras
17B01 Identities, free Lie (super)algebras

Citations:

Zbl 0996.08008
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References:

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