Prepositions (P) area closed, "unpredictable," often grammaticized, and highly idiomatic word class
Prepositional phrases (NP) typically serve as postmodifiers of nouns, NPs, and adjectives; as subject and object complements, and as independent "sentence adjuncts."
The prepositional phrase can be analyzed as consisting of three elements: an optional Pre-Modifier (usually an "intensifier," Head, and Complement. The complement is obligatory; prepositions don't appear in isolation.
Some common premodifiers: "very much," "hardly," "especially," and other intensifying words and phrases – typically adverbs, but their function is more important than their class: very much in love, hardly at ease, especially under the weather, infinitely in your debt, securely on top
Complements of prepositions are most commonly of two general types:
Noun phrases: in a box, a pickle, the back of a pickup, the United States, your dreams, ten years, the time it takes to thread a needle; about his age, Jeff's height, the amount I wanted, that time of night; by hand, the waters of Babylon, Steve's house, the time I get to Phoenix
clauses with -ing participle or wh- word: after causing a fuss, getting up in the morning, swimming the English Channel, dying, cooking Whisper his breakfast; about what you said to me, what I saw, when he graduated, how we figured it, what a pickup can carry
Grammaticized uses of prepositions often correspond to cases of nouns in more highly inflected languages: "by" with the agent of passive verbs, "of" genitives, "to" datives (as IO), "with" expressing means, "as" and "than" with comparative expressions.
Certain verbs take idiomatic prepositions: account, beg, ask, stand, wait "for"; believe, engage, confide, result "in"; etc.