Chain

In syntax, a chain is a set of syntactic elements subject to specific conditions.

Formally:

(a1,...,an), 1 =< n, is a chain iff (i) every a has the same subscript, i.e       (a1,...,an) = (aj1,...,ajn) (ii) for every i < n, a1 antecedent-governs ai+1

Given this definition a1 is called the head of the chain, an the foot, and each pair (a1,ai+1) is a link. The superscripts only serve to distinguish elements which are otherwise identical; so, superscripts are left out in the notation if the elements are not identical. Different kinds of chains are distinguished. A chain is an A-chain if a1 is in an A-position, and an A-bar-chain if a1 is in an A-bar position.

Examples
In (i) both (Johni, ti) and (the carj) are A-chains. The chain (Johni, ti) consists of one link, Johni being the head and ti being the foot. The chain (the carj) has no link, and the carj is both its head and its foot.

(i) Johni was hit ti by the carj (ii) Whoi ti1 seems ti2 to have been hit ti3 by the carj

In (ii), the chain (whoi, ti1, ti2, ti3) is an A-bar-chain, since the head whoi is in an A-bar-position. The foot (i.e. ti3) of this chain is theta-marked (by hit). The element t1 is case marked (by the matrix INFL). Hence, the chain satisfies the case filter and the theta criterion. The A-chain (ti1, ti2) is an example of a non-maximal chain, since this chain, being part of the maximal chain (whoi, ti1, ti2, ti3) contains no theta-position.

Comment
Grammatical properties, such as theta-roles and Case visibility (visibility condition) are properties of maximal chains. A chain is maximal if it contains a theta-position. In general, maximal chains are simply called chains.

Link
Utrecht Lexicon of Linguistics