▁first ▁implies ▁x yz ▁= ▁x y ( z x z ) ▁= ▁( x ( yz ) x ) z ▁= ▁x z . ▁ ▁There ▁is ▁a ▁complete ▁classification ▁of ▁rect angular ▁bands . ▁ ▁Given ▁arbitrary ▁sets ▁I ▁and ▁J ▁one ▁can ▁define ▁a ▁sem igr oup ▁operation ▁on ▁I   ×   J ▁by ▁setting ▁▁▁▁ ▁The ▁resulting ▁sem igr oup ▁is ▁a ▁rect angular ▁band ▁because ▁▁ ▁for ▁any ▁pair ▁( i ,   j ) ▁we ▁have ▁( i ,   j )   ·   ( i ,   j )   =   ( i ,   j ) ▁ ▁for ▁any ▁two ▁pairs ▁( ix ,   j x ), ▁( iy ,   j y ) ▁we ▁have ▁▁▁ ▁In ▁fact , ▁any ▁rect angular ▁band ▁is ▁is omorphic ▁to ▁one ▁of ▁the ▁above ▁form . ▁ ▁Left - zero ▁and ▁right - zero ▁bands ▁are ▁rect angular ▁bands , ▁and ▁in ▁fact ▁every ▁rect angular ▁band ▁is ▁is omorphic ▁to ▁a ▁direct ▁product ▁of ▁a ▁left - zero ▁band ▁and ▁a ▁right - zero ▁band . ▁All ▁rect angular ▁bands ▁of ▁prime ▁order ▁are ▁zero ▁bands , ▁either ▁left ▁or ▁right . ▁A ▁rect angular ▁band ▁is ▁said ▁to ▁be ▁purely ▁rect angular ▁if ▁it ▁is ▁not ▁a ▁left - zero ▁or ▁right - zero ▁band . ▁ ▁In ▁categor ical ▁language , ▁one ▁can ▁say ▁that ▁the ▁category ▁of ▁non empty ▁rect angular ▁bands ▁is ▁equivalent ▁to ▁, ▁where ▁ ▁is