- regular ▁bands , ▁right - regular ▁bands ▁and ▁regular ▁bands , ▁specific ▁sub classes ▁of ▁bands ▁that ▁lie ▁near ▁the ▁bottom ▁of ▁this ▁lattice , ▁are ▁of ▁particular ▁interest ▁and ▁are ▁briefly ▁described ▁below . ▁ ▁Vari eties ▁of ▁bands ▁A ▁class ▁of ▁bands ▁forms ▁a ▁variety ▁if ▁it ▁is ▁closed ▁under ▁formation ▁of ▁subs em igr oups , ▁hom omorphic ▁images ▁and ▁direct ▁product . ▁Each ▁variety ▁of ▁bands ▁can ▁be ▁defined ▁by ▁a ▁single ▁ ▁defining ▁identity . ▁ ▁Sem il att ices ▁Sem il att ices ▁are ▁exactly ▁commut ative ▁bands ; ▁that ▁is , ▁they ▁are ▁the ▁bands ▁satisfying ▁the ▁equation ▁ ▁x y ▁= ▁y x ▁for ▁all ▁x ▁and ▁y . ▁ ▁Zero ▁bands ▁A ▁left - zero ▁band ▁is ▁a ▁band ▁satisfying ▁the ▁equation ▁ ▁x y   =   x , ▁wh ence ▁its ▁C ay ley ▁table ▁has ▁constant ▁rows . ▁ ▁Sym met r ically , ▁a ▁right - zero ▁band ▁is ▁one ▁satisfying ▁ ▁x y   =   y , ▁so ▁that ▁the ▁C ay ley ▁table ▁has ▁constant ▁columns . ▁ ▁Rect angular ▁bands ▁A ▁rect angular ▁band ▁is ▁a ▁band ▁S ▁that ▁satisfies ▁ ▁x y x ▁= ▁x ▁for ▁all ▁x ,   y   ∈   S , ▁or ▁equival ently , ▁x yz   =   x z ▁for ▁all ▁x ,   y ,   z   ∈   S , ▁The ▁second ▁character ization ▁clearly ▁implies ▁the ▁first , ▁and ▁convers ely ▁the