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I was having a look at the block product principle for finite monoids. I wanted to see the derivation of LTT = Acom$*$LI, using an example. But I can't come up with a non-trivial example of a monoid in LI. Any help is appreciated.

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There is no nontrivial monoid in LI. LI is actually a (pseudo)-variety of finite semigroups: it is the class of finite semigroups $S$ such that, for every idempotent $e \in S$, $eSe = e$. If $S$ is a monoid, you can take $e = 1$, the identity of the monoid and get $S = 1S1 = 1$ so the monoid has to be trivial.

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  • $\begingroup$ Thanks! I had noticed my mistake. Also, just out of curiosity, is there an extension of block product principle for other classes of semigroups and simultaneously for larger classes of logic? P.S. I still can't believe I was answered by J.E. Pin! $\endgroup$ – Baby_Faced_Assassin Sep 11 at 18:07
  • $\begingroup$ Well, if you are satisfied with the answer, don't forget you have the possibility to accept it. About your P.S., you should ask a separate question, but perhaps on math.stackexchange, since it looks like a purely mathematical question. $\endgroup$ – J.-E. Pin Sep 12 at 10:08
  • $\begingroup$ Oh I didn't know how to accept the solution. I am fairly new to using stackexchange... And yes, I shall first think about it a bit more before asking it officially. $\endgroup$ – Baby_Faced_Assassin Sep 12 at 16:32

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