Many PPs exhibit some amount of lexicalization or idiomaticity. This is especially true of PPs that tend to be used predicatively. In general it is extremely difficult to establish tests to distinguish idiomatic PPs from fully productive combinations. However, the usual criteria apply for the supersense analysis.
See further discussion at Characteristic.
A PP idiom is a fixed or semi-fixed expression consisting of an adposition plus its complement (usually an NP, AdjP, or AdvP), which must be a complete phrase. In some of these expressions the complement may take variable modifiers (e.g., on_ ONE's _own: see `$). The PP idiom as a whole does not take a complement (is intransitive). A fixed expression ending in a transitive preposition like of or as (in_search_of, as_long_as) requires a complement, and thus is not a PP idiom.1
Certain idiomatic constructions involve a preposition that requires a reflexive direct object.
PERFORM-ACTIVITY for oneself.
PERFORM-ACTIVITY by oneself.
BE by oneself.