I think it's clear that commutative semigroups S that are also bands, i.e. $e^2 = e$ for all e, correspond to finite posets (consider the elements of the semigroups as sets, where the intersection of two sets is their product), satisfying the condition that there is a unique minimal element in this finite poset (to make sure multiplication is well-defined), and also satisfying the condition that there is a unique maximal element in this semigroup if we want it to have an identity. (Unless my reasoning is wrong?)

I was wondering if there is a standard representation theory for such semigroups covered in a paper somewhere. I've thought about it a bit and reduced it to a combinatorial problem, but I'd rather not spend time on it since it is almost certainly a well-known (and easy probably) result.

Is there a more general representation theory for bands available? What about representation theory of rectangular bands? (which I presume is slightly more challenging, unless the problem above grows much more difficult as the poset grows more complicated, which is also a possibility).