Complement

In syntax, the term complement is sometimes used in the same sense as argument or, more precisely, argument term. A constituent A is the complement of another constituent B iff A is required by the valency of B. Mostly, the term 'complement' is only used for internal, but not for external arguments, i.e. the subject of a given predicate is regarded as an argument, but not as a complement of that predicate.

Examples
YP in [H' H YP], where H is a head and H' the projection of H. According to X-bar theory, the complement of a head X0 is defined either as a position attached or adjoined to X', or as a sister of X0. Thus, in the configuration, either A and B are complements of X0, or just B (the sister of X0).

X'              / | /	|                 A	X' /   |           /	|          B	X0

Sometimes, the complement of a head X0 is equated with its internal argument(s).

(1) Mary is a beautiful girl.

(2) The demonstration was on Sunday.

(3) *The demonstration was

Comments
The term is fairly common in the syntax of German (Ergänzung), especially in German dependency / valence grammar (e.g. Vater 1978, Engel 1988).

In English-language linguistics the term complement in this sense is common only as part of the term complement clause (= "argument clause", a clause that is an argument).

The term has been employed only in relation to nominal or adjectival expressions which combine with the copula. In traditional grammar, the term is used to refer to any word or phrase (other than the verb itself) which is an obligatory constituent of the predicate (Lyons 1968: 345).

The difference between an adjunct and a complement is, the former is an optional (extrnuclear) constituent, and the latter an obligatory (nuclear) constituent of the sentence.

Polysemy
The term complement also refers to
 * a syntactic position in the X-bar schema (see complement (in X-bar theory)).
 * a non-subject argument, i.e. roughly an object. (This usage seems to have been widespread in 20th century traditional grammar. In this usage, a sentence consists of subject and predicate, and the predicate consists of the verb plus the complements and optional adjuncts. This meaning is standard in Russian-language linguistics.)
 * a complement clause (since complement in the sense 'argument' is much rarer than complement clause, this can be abbreviated to complement without serious risk of misunderstanding in many contexts; see Rosenbaum for an influential work)
 * a predicate nominal (see complement (predicative))

Synonyms

 * argument
 * actant
 * term (in relational grammar)

Origin
The term is not old in English (the OED's first attestation is from 1874),and its equivalents are much more common in French, German (Ergänzung) and Russian (dopolnenie).

The oldest meaning of complement seems to have been dependent (i.e. not just arguments, but also adjuncts), cf. Grande encyclopédie du XIX siècle (1868) "compléments: mots qui servent à compléter l'idée exprimée par d'autres mots" (the examples make it clear that all kinds of dependents are meant).

Link
Utrecht Lexicon of Linguistics

other languages

 * Chinese 补语
 * German Ergänzung, Komplement
 * Russian Дополнение (Dopolnenie)