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.
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.