xy b   =   ay x b ▁for ▁all ▁a , ▁b , ▁x , ▁and ▁y   ∈   S . ▁ ▁Left - regular ▁bands ▁ ▁A ▁left - regular ▁band ▁is ▁a ▁band ▁S ▁satisfying ▁▁ ▁x y x ▁= ▁x y ▁for ▁all ▁x ,   y ▁  ∈   S ▁ ▁If ▁we ▁take ▁a ▁sem igr oup ▁and ▁define ▁a ▁ ≤ ▁b ▁ ▁if ▁and ▁only ▁if ▁ab ▁= ▁b , ▁we ▁obtain ▁a ▁partial ▁ordering ▁if ▁and ▁only ▁if ▁this ▁sem igr oup ▁is ▁a ▁left - regular ▁band . ▁ ▁Left - regular ▁bands ▁thus ▁show ▁up ▁naturally ▁in ▁the ▁study ▁of ▁pos ets . ▁ ▁Right - regular ▁bands ▁ ▁A ▁right - regular ▁band ▁is ▁a ▁band ▁S ▁satisfying ▁▁ ▁x y x ▁= ▁y x ▁for ▁all ▁x ,   y ▁  ∈   S ▁ ▁Any ▁right - regular ▁band ▁becomes ▁a ▁left - regular ▁band ▁using ▁the ▁opposite ▁product . ▁ ▁Indeed , ▁every ▁variety ▁of ▁bands ▁has ▁an ▁' opp os ite ' ▁version ; ▁this ▁gives ▁rise ▁to ▁the ▁reflection ▁symmetry ▁in ▁the ▁figure ▁below . ▁ ▁Reg ular ▁bands ▁A ▁regular ▁band ▁is ▁a ▁band ▁S ▁satisfying ▁ ▁z x zy z ▁= ▁z xyz ▁for ▁all ▁x ,   y ,   z   ∈   S ▁ ▁L attice ▁of ▁vari eties ▁ ▁When ▁partially ▁ordered ▁by ▁inclusion , ▁vari eties ▁of ▁bands ▁naturally ▁form ▁a ▁lattice , ▁in ▁which ▁the ▁meet ▁of