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For a distance fixed prime numbers $p$ and $q$, let $G_p$, $G_q$ and $Sl$ the pseudovarieties of all finite $p$-group, $q$-group and semilattices respectively. Dose the following equality hold? $Sl*G_p*G_q=(Sl*G_p)\mal G_q$

where $*$ denote the semidirect product and $\mal$ the mal'cev product of pseudovarieties

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    $\begingroup$ The answer is probably no but I don't recall if explicit examples have been constructed. Using the Higgins-Margolis construction one can show Sl*G_p is not local. So there is no reason equality should hold. You might contact Karl Auinger. He used to think about such things and might remember what is known $\endgroup$
    – Benjamin Steinberg
    Apr 8, 2015 at 18:11
  • $\begingroup$ Dear Steinberg thanks. I sow the Higgins-Margolis construction, but I can not get form that why for every proper pseudovariety of groups $H$, $Sl*H$ is not local. $\endgroup$
    – user182085
    Apr 9, 2015 at 15:21
  • $\begingroup$ Auinger modified the construction to get this. I forgot the details but ask him. This was 10 years ago $\endgroup$
    – Benjamin Steinberg
    Apr 9, 2015 at 15:42
  • $\begingroup$ I think Auinger got infinite vertex rank. I don't think he published it $\endgroup$
    – Benjamin Steinberg
    Apr 9, 2015 at 15:45
  • $\begingroup$ What do you mean by the infinite vertex rank? $\endgroup$
    – user182085
    Apr 9, 2015 at 16:01

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