 There is a complete classification of rectangular bands.Given arbitrary sets "I" and "J" one can define a semigroup operation on "I" × "J" by setting

The resulting semigroup is a rectangular band because 

In fact, any rectangular band is isomorphic to one of the above form.Left-zero and right-zero bands are rectangular bands, and in fact every rectangular band is isomorphic to a direct product of a left-zero band and a right-zero band.