# Dominance

Dominance is a dominance is a binary relation between nodes in a tree structure which can be defined as follows:

```(i) Node A dominates node B iff A is higher up the tree than B such that you
can trace a line from A to B going only downwards
```

The Dominance relation has the following logical properties:

``` - irreflexivity:	a node does not dominate itself
- asymmetry:		if A dominates B, B does not
dominate A
- transitivity:	if A dominates B, and B dominates C,
then A dominates C
```

A relation which has these three properties is called a partial order.

### Example

node A dominates all other nodes in (ii). C dominates F and G, but F and G do not dominate C.

```(ii)		 A
/ \
/   \
B	    C
/ \   / \
D  E   F G
```

Also, node C in (ii) does not dominate the nodes B, D, or E, nor is it dominated by either of these nodes. Furthermore, node A immediately dominates the nodes B and C: there is no intervening node between A on the one hand and B and C on the other hand, i.e. there is no node N such that N dominates B and C and is dominated by A. Analogously, D and E are immediately dominated by B; F and G by C. Nodes D and E are called sisternodes. The same holds for F and G, and B and C. The nodes B and C are daughters of A; D and E are daughters of B; F and G daughters of C. In recent literature (May 1985, Chomsky 1986b), the dominance relation has been redefined in terms of the notion segment:

```	(iii) A is dominated by B iff A is dominated by
every segment of B
```

In (iv), there are two nodes XP1 and XP2, the result of movement by adjunction, which are each taken to be a segment of the maximal projection XP (= {XP2, XP1}) of X.

```(iv)		XP2
/|
/	|
ZPi	XP1
|
X'
|\
| \
X  YP
|
Y'
|\
| \
Y  ZP
|
ti
```

XP1 is the original maximal projection. It is called the minimal maximal projection or the base maximal projection. The adjoined ZP is dominated in the original sense of (i) by the topmost maximal projection segment XP2 but it is not dominated in the sense of (i) by the minimal maximal projection segment XP1. Thus, by the definition in (iii), ZP is not dominated by XP. ZP is not fully part of the projection of X: it is not dominated by every segment of the maximal projection of X. YP, in contrast, is dominated both by XP1 and by XP2. YP, then, is completely inside the projection of X, i.e. YP is dominated by XP in the sense of (iii). Even though ZP is not dominated by the maximal projection of X, it is not entirely outside the maximal projection of X, being dominated by the topmost node XP2. It is said then, that ZP is not excluded from XP. Exclusion is defined in (v).

```(v) Exclusion
B excludes A if no segment of B dominates A
```

Chomsky (1992) distinguishes s(egment)-domination from c(ategory)-domination. A c-dominates B if every segment of A dominates B. A s-dominates B if some segment of A dominates B.