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Let $G$ be a group and $V=Var(G)$ be the variety generated by $G$. Suppose the axiomatic rank of $V$ is $n$. Let $Id(V)$ be the set of all identities of $V$.

1- Can we say that every element of $Id(V)$ is equivalent to some identity of the form $$ w(x_1, \ldots, x_n)\approx 1?$$

2- Let $G$ be finite. Can we prove that the axiomatic rank of $V$ is finite without using the fact that $V$ is finitely based?

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    $\begingroup$ What is axiomatic rank? Number of identities or variables? $\endgroup$
    – Benjamin Steinberg
    Commented Oct 7, 2014 at 20:36
  • $\begingroup$ Minimum number of variables. $\endgroup$
    – Sh.M1972
    Commented Oct 8, 2014 at 3:23
  • $\begingroup$ For question 1 what do you mean by equivalent. $\endgroup$
    – Benjamin Steinberg
    Commented Oct 8, 2014 at 3:46
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    $\begingroup$ For question 2 Mark Sapir proved a finite semigroup is finitely based iff it has finite axiomatic rank in your sense. He may have used that all finite groups are finitely based though. So most likely the answer is no. $\endgroup$
    – Benjamin Steinberg
    Commented Oct 8, 2014 at 3:53
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    $\begingroup$ The trivial variety has axiomatic rank 1, right? for $n\geq 2$, the free group $F_m$ embeds in $F_n$, this convince me that there may be a way of expressing an $m$-variable identity in terms of $n$ variables. $\endgroup$
    – Sh.M1972
    Commented Oct 9, 2014 at 6:46

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Concerning question 2: by a result of Birkhoff, a finite algebra (of finite type) is finitely based if and only if it is of finite axiomatic rank (see e.g. Thm 4.2. in the book of Burris-Sankappanavar). So the question essentially asks for a new proof of the Oates-Powell-Theorem.

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  • $\begingroup$ Thank you for the reference. Just can't understand one point: the theorem of Birkhoff concerns all finite algebras of finite type and as I see it is published 1935. In the comments Benjamin Steinberg says that it is proved for finite semigroups by Mark Sapir (may be on 1988). I can't download the paper of Sapir, so I don't know actually what he did. $\endgroup$
    – Sh.M1972
    Commented Oct 11, 2014 at 2:46

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