mail order vs purchase order

mail order and purchase order both are nouns.

Word	Noun	Adjective	Verb	Adverb
mail order	Yes	No	No	No
purchase order	Yes	No	No	No

As nouns, purchase order is a hypernym of mail order; that is, purchase order is a word with a broader meaning than mail order:

mail order: a purchase negotiated by mail
purchase order: a commercial document used to request someone to supply something in return for payment and providing specifications and quantities

Other hypernyms of mail order include order.

mail order (noun)	purchase order (noun)
a purchase negotiated by mail	a commercial document used to request someone to supply something in return for payment and providing specifications and quantities