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Let $\mathcal{L}$ be a finite first-order language with no relation symbols, and $\mathcal{K}:=\mathcal{V}(\Theta)$ a variety in this language definited by a set of identities $\Theta$.

My questions are motivated, of course, by Noether's Theorem in invariant theory, and Chevalley-Shephard-Todd theorem (which can be seem as statements in the variety of commutative associative algebras) for algebraic systems (using Malcev terminology).

Question 1 Let $A$ be a finitely generated algebraic system in $\mathcal{K}$, and $G$ a finite group acting by $\mathcal{K}$-automorphisms on $A$. When is $A^G:=\{a \in A| g.a=a, \forall g \in G\}$, again finitely generated? A subcase of important interest is when $A$ is a relatively free algebraic system of $\mathcal{K}$.

Question 2 Let $A$ be a finitely generated and relatively free algebraic system in $\mathcal{K}$ and $G$ be a finite group of automorphisms as above. When is $A^G$ is a (possibly infinitely generated) relatively free algebraic system of $\mathcal{K}$ itself?

For varieties of associative algebras the work on Questions 1 and 2 has reached a very mature form (cf. Formanek, Noncommutative invariant theory, MR0810646). There has been an intensive study of these questions for varieties of Lie algebras by a number of people (V. Drenksy, V. Petrogradskii, etc) but I know of no work for other important variaties of non-associative algebras (such as Jordan algebras, alternative algebras, Malcev algebras, etc), and also nothing about general algebraic systems.

Refinement of Questions 1 and 2 Is there any general result in Universal algebra (or Model theory) that is relevant for the pourposes of these questions? Is something known about variaties of non-associative algebras (other than Lie)?

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    $\begingroup$ Can you please explain what do you mean by the symbol $A^G$? $\endgroup$ – Jakub Opršal Jul 21 at 15:03
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    $\begingroup$ @JakubOpršal $A^G = \{ a \in A| g a = a, \forall g \in G \}$. Thanks, I will edit the question for clarity $\endgroup$ – jg1896 Jul 21 at 15:33

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