▁top ▁element . ▁ ▁Any ▁ordinary ▁mon oid ▁ ▁is ▁a ▁small ▁mon oid al ▁category ▁with ▁object ▁set ▁, ▁only ▁ident ities ▁for ▁morph isms , ▁ ▁as ▁tensor product ▁and ▁ ▁as ▁its ▁identity ▁object . ▁Con vers ely , ▁the ▁set ▁of ▁isomorphism ▁classes ▁( if ▁such ▁a ▁thing ▁makes ▁sense ) ▁of ▁a ▁mon oid al ▁category ▁is ▁a ▁mon oid ▁w . r . t . ▁the ▁tensor ▁product . ▁ ▁Mon oid al ▁pre orders ▁ ▁Mon oid al ▁pre orders , ▁also ▁known ▁as ▁" pre ordered ▁mon oid s ", ▁are ▁special ▁cases ▁of ▁mon oid al ▁categories . ▁This ▁sort ▁of ▁structure ▁comes ▁up ▁in ▁the ▁theory ▁of ▁string ▁re writing ▁systems , ▁but ▁it ▁is ▁pl ent iful ▁in ▁pure ▁mathematics ▁as ▁well . ▁For ▁example , ▁the ▁set ▁ ▁of ▁natural ▁numbers ▁has ▁both ▁a ▁mon oid ▁structure ▁( using ▁+ ▁and ▁ 0 ) ▁and ▁a ▁pre order ▁structure ▁( using ▁ ≤ ), ▁which ▁together ▁form ▁a ▁mon oid al ▁pre order , ▁basically ▁because ▁ ▁and ▁ ▁implies ▁. ▁We ▁now ▁present ▁the ▁general ▁case . ▁ ▁It ' s ▁well ▁known ▁that ▁a ▁pre order ▁can ▁be ▁considered ▁as ▁a ▁category ▁C , ▁such ▁that ▁for ▁every ▁two ▁objects ▁, ▁there ▁exists ▁at ▁most ▁one ▁morph ism ▁ ▁in ▁C . ▁If ▁there ▁happens ▁to ▁be ▁a ▁morph ism ▁from ▁c ▁to ▁c ' ▁, ▁we ▁could ▁write ▁, ▁but ▁in ▁the ▁current ▁section ▁we ▁find ▁it ▁more