▁the ▁category ▁with ▁non empty ▁sets ▁as ▁objects ▁and ▁functions ▁as ▁morph isms . ▁ ▁This ▁implies ▁not ▁only ▁that ▁every ▁non empty ▁rect angular ▁band ▁is ▁is omorphic ▁to ▁one ▁coming ▁from ▁a ▁pair ▁of ▁sets , ▁but ▁also ▁these ▁sets ▁are ▁uniqu ely ▁determined ▁up ▁to ▁a ▁canonical ▁isomorphism , ▁and ▁all ▁hom omorph isms ▁between ▁bands ▁come ▁from ▁pairs ▁of ▁functions ▁between ▁sets . ▁ ▁If ▁the ▁set ▁I ▁is ▁empty ▁in ▁the ▁above ▁result , ▁the ▁rect angular ▁band ▁I   ×   J ▁is ▁independent ▁of ▁J , ▁and ▁vice ▁vers a . ▁ ▁This ▁is ▁why ▁the ▁above ▁result ▁only ▁gives ▁an ▁equivalence ▁between ▁non empty ▁rect angular ▁bands ▁and ▁pairs ▁of ▁non empty ▁sets . ▁ ▁Rect angular ▁bands ▁are ▁also ▁the ▁T - al gebras , ▁where ▁T ▁is ▁the ▁mon ad ▁on ▁Set ▁with ▁T ( X )= X × X , ▁T ( f )= f × f , ▁ ▁being ▁the ▁diagonal ▁map ▁, ▁and ▁. ▁ ▁Normal ▁bands ▁A ▁normal ▁band ▁is ▁a ▁band ▁S ▁satisfying ▁ ▁z xyz   =   zy x z ▁for ▁all ▁x , ▁y , ▁and ▁z   ∈   S . ▁This ▁is ▁the ▁same ▁equation ▁used ▁to ▁define ▁med ial ▁mag mas , ▁and ▁so ▁a ▁normal ▁band ▁may ▁also ▁be ▁called ▁a ▁med ial ▁band , ▁and ▁normal ▁bands ▁are ▁examples ▁of ▁med ial ▁mag mas . ▁We ▁can ▁also ▁say ▁ ▁a ▁normal ▁band ▁is ▁a ▁band ▁S ▁satisfying ▁ ▁a