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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

user182085
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    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 – Benjamin Steinberg Apr 08 '15 at 18:11
  • 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. – user182085 Apr 09 '15 at 15:21
  • Auinger modified the construction to get this. I forgot the details but ask him. This was 10 years ago – Benjamin Steinberg Apr 09 '15 at 15:42
  • I think Auinger got infinite vertex rank. I don't think he published it – Benjamin Steinberg Apr 09 '15 at 15:45
  • What do you mean by the infinite vertex rank? – user182085 Apr 09 '15 at 16:01
  • The global cannot be defined by pseudo identities on graphs with a bounded number of vertices – Benjamin Steinberg Apr 09 '15 at 16:08
  • But by construction of a basis in [Almeida-Azevedo-Teixeira'2000], for pseudovarity $gH$ which $H$ contains Brandt semigroup, $gH$ is of finite vertex rank. – user182085 Apr 09 '15 at 16:18
  • I don't believe so. There are plenty of pseudovarieties of infinite vertex rank containing Brandt like A\ast G. If the pseudovariety is infinitely based using infinitely many variables then the construction of the paper you cite uses an unbounded number of vertices. The methods of Cowan, Reilly and volkov, generalizing Higgins Marsalis shows Sl*V is not finitely based for V a proper nontrivial pseudovariety of groups. – Benjamin Steinberg Apr 09 '15 at 17:14
  • But I am working with the finitely many variables. – user182085 Apr 10 '15 at 10:07
  • Thanks for your time. I got why it is not local. – user182085 Apr 10 '15 at 12:00
  • You cannot define Sl*H on finitely many variables by the results of Cowan et AL. – Benjamin Steinberg Apr 10 '15 at 16:31

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