Let $V$ and $W$ be pseudovarieties of finite groups. For a finite inverse monoid $M$, the $V$-kernel of $M$ is defined to be the intersection of all sets $f^{−1}(1)$, $f$ is a relational morphism from $M$ into a finite group $G\in V$, and denote it by $K_V(M)$. We always have $K_V(K_W(M))\subseteq K_{V*W}(M)$ where $V*W$ is the semidirect product of pseudovarieties. My question is ''when the reverse inclusion holds?''
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$\begingroup$ My guess is that if W is locally finite then equality holds. Probably it fails in general. $\endgroup$– Benjamin SteinbergCommented Apr 7, 2015 at 17:49
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$\begingroup$ Dear Benjamin How could you guess if $W$ is locally finite then equality holds? $\endgroup$– user182085Commented Apr 23, 2015 at 15:06
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$\begingroup$ Maybe it is not true but It's my feeling $\endgroup$– Benjamin SteinbergCommented Apr 23, 2015 at 19:21
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$\begingroup$ If your guess is true then the equality $Sl*V*W=(Sl*V)\malcev W$ holds. I think it is weird. $\endgroup$– user182085Commented Apr 24, 2015 at 10:26
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$\begingroup$ Some people found J*G=JmG weird. Perhaps I want V closed under extension and W locally finite $\endgroup$– Benjamin SteinbergCommented Apr 24, 2015 at 13:53
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