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A set $\Sigma$ of group identities is called bounded if there is $n\geq 1$ such that for any $(w\approx 1)\in \Sigma$, we have $w\in F(x_1, \ldots, x_n)$. A variety $\mathbf{V}$ is called bounded defined if $\mathbf{V}=Mod(\Sigma)$ for some bounded set $\Sigma$.

Question: Is there an example of a bounded defined non-finitely based variety of groups?

P.S. $F(x_1, \ldots, x_n)$ is the free group of rank $n$.

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  • $\begingroup$ I think there is a better way to ask this question. Let T be a "narrow" equational theory, with closure of T logically equivalent to closure of S, where S is an equational basis in group theory at most n distinct variables. Does cl(T) have a finite basis necessarily? $\endgroup$
    – The Masked Avenger
    Jan 19, 2014 at 4:29

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Yes. It is usually called a non-finitely based variety of finite axiomatic rank.

Actually, one of the first examples of non-finitely based variety of groups has axiomatic rank two; Adian proved that the following set of identities is independent: $$ \{[x^{pn},y^{pn}]^n=1\;|\; p \hbox{ is prime}\}, $$ where $n\ge 1003$ is any given odd number. Independence means that none of the laws is a consequence of the other ones.

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  • $\begingroup$ Dear Anton Klyachko, thank you for the example. Would you please check your emails? I sent a message. $\endgroup$
    – Sh.M1972
    Jan 19, 2014 at 16:46
  • $\begingroup$ Sorry for this question, it may be comic: the example of Adian has axiomatic rank 2 which is smaller than the axiomatic rank of the variety of groups. can you give an example of axiomatic rank >2? $\endgroup$
    – Sh.M1972
    Jan 19, 2014 at 20:14

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