# Example of a locally trivial semigroup

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.