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SYNTAX: C-command is a binary relation between nodes in a tree structure which is defined as follows:

(i) Node A c-commands node B iff

    a  A =/= B,
    b  A does not  dominate B and B does not 
       dominate A, and
    c  every X that dominates A also dominates B.

(ii)		X2
		/ \
	         A  X1
                    / \
		 B  C
In (ii) A c-commands B since A =/= B (cf. (i)a), A does not dominate B, nor does B dominate A (cf. (i)b); and the node which dominates A, X2, also dominates B (cf. (i)c). X1 in (ii) is not relevant to (i)c: although it dominates B, it does not dominate A.
For the possible choices of X in (i)c several options have been proposed. The first option is to interpret X as any branching node. Under this interpretation A c-commands B iff (ia) and (ib) are met and the first branching node dominating A also dominates B. This structural relation is sometimes referred to as strict c-command. EXAMPLE: In (iii) V c-commands NP the book, but not the PP in the store: when we start from V and move upwards, the first branching node we reach is V'1. This node dominates the NP the book and it does not dominate the PP in the store. By the same token the P in c-commands the NP the store; it does not c-command the V'1 buy the book, nor the V and the NP contained in V'1, because the first branching node dominating P does not dominate V'1.
(iii)	 VP
	 | PP
	 |   \
	 |    P'
	 |    |\
	 V'1 | \	
	 |\   P  NP
	 | \  |   |
	 |  \ in the store
	 |   \
	 V   NP 
   	 |   |	
	buy  the book 
An alternative option for the possible values of X in (i)c is to count only maximal projections. Under this interpretation A is said to m-command B.
EXAMPLE: V in (iii) m-commands both the NP the book and the PP in the store and what is contained in them. P does not m-command V, because there is a maximal projection PP which dominates P and does not dominate V, nor the NP the book.
The minimal phrase which contains a c- or m-commanding element A is the c- or m-command domain of that element. The notion 'minimal phrase' is defined according to the interpretation of X in the definition in (i). Thus, if A m-commands B, the minimal phrase containing A is labeled XP. The m-command domain, then, is the smallest maximal projection containing A. In (iii) PP, not VP, is the m-command domain of P, since PP is the smallest maximal projection in which P appears. If A c-commands B the minimal phrase is the first branching node dominating A. Thus, V'1 in (iii) is the c-command domain of V. The c-command domain of an element must be a constituent, given that it consists of all the material dominated by one node; hence the term c(onstituent)-command. Other proposals restrict X to lexical categories, major categories. The notion c-command plays a role in the definitions of government, binding, and scope.
LIT. Aoun, Y. & D. Sportiche (1983)
Chomsky, N. (1986b)
Chomsky, N. (1986a)
Chomsky, N. (1981)
Reinhart, T. (1976)