all spanning intervals where i > j or k > l represent the empty set of intervals, because the range of possible endpoints would be empty. Fifth, all spanning intervals where i = j and either  or Æ is false (indicating an open range for the lower endpoint) represent the empty set of intervals, because the range of possible endpoints would be empty. Similarly, all spanning intervals where k = l and either  or  is false (indicating an open range for the upper endpoint) also represent the empty set of intervals. Sixth, all spanning intervals where i = l and either  or  is false (indicating an open interval) also represent the empty set of intervals, because the endpoints of an open interval must be different. Seventh, if j = l and  is false, the value of Æ does not affect the denotation, because if j = l and  is false, the upper endpoint must be less than l and the lower endpoint must be less than or equal to j which equals l, so the lower endpoint must be less than j. Similarly, if k = i and  is false, the value of  does not affect the denotation. Eighth, if j = l and either  or  is false, the value of Æ does not affect the denotation, because the lower endpoint of an open interval must be less than its upper endpoint. Similarly, if k = i and either  or  is false, the value of  does not affect the denotation.

To create a situation where the extension of every spanning interval has a unique representation, let us represent all empty sets of intervals as fg. When the values of i, j, k, l, , , , Æ, , or can be changed without changing the denotation, we will select the tightest such values. In other words, false values for the Boolean parameters, maximal values for the lower bounds, and minimal values for the upper bounds. Thus whenever we represent a spanning interval [
 [i; j℄Æ ; [k; l℄ ℄

explicitly, it will have a nonempty extension and will satisfy the following normalization criterion:

$$\begin{array}{l}{{(i\neq\infty)\wedge(j\neq-\infty)\wedge(k\neq\infty)\wedge(l\neq-\infty)\wedge}}\\ {{\left[(i=-\infty)\to-\gamma\right]\wedge\left[(j=\infty)\to-\delta\right]\wedge\left[(k=-\infty)\to-\epsilon\right]\wedge\left[(l=\infty)\to-\gamma\right]\wedge(j\leq l)\wedge(k\geq i)\wedge}}\\ {{(i<i)\wedge(k\leq l)\wedge}}\end{array}$$

$\mathbf{b}$
(4) (i  j) ^ (k  l)^

(5) [(i 6= j) _ (
 ^ Æ)℄ ^ [(k 6= l) _ ( ^  )℄^

$$\begin{array}{l}{{[(l^{\alpha}\gamma^{\epsilon})\vee(l^{\alpha}\gamma^{\beta})]^{\prime}}}\\ {{\{[(j=l)\wedge\neg\zeta]\rightarrow\neg\delta\}\wedge\{[(k=i)\wedge\neg\gamma]\rightarrow\neg\epsilon\}\wedge}}\\ {{\{[(j=l)\wedge(\neg\alpha\vee\neg\beta)]\rightarrow\neg\delta\}\wedge\{[(k=i)\wedge(\neg\alpha\vee\neg\beta)]\rightarrow\neg\epsilon\}}}\end{array}$$

Criteria (1) through (8) correspond to points one through eight above.

A spanning interval [
 [i; j℄Æ ; [k; l℄ ℄ is normalized if i, j, k, l, , , , Æ, , and  cannot be changed without changing its denotation. Given a (potentially non-normalized) spanning interval i, its normalization hii is the smallest set of normalized spanning intervals that represents the extension

