One can extend the definition of Allen's relations to both open and closed intervals as follows.

The relation i1 = i2 holds if the corresponding endpoints of i1 and i2 are equal and have the same openness. The relation i1 < i2 holds if the right endpoint of i1 precedes the left endpoint of i2 or if they are equal and both open. For example, [1; 3℄ < [4; 5℄ and [1; 3) < (3; 5℄, but [1; 3℄ 6< (3; 5℄,
[1; 3) 6< [3; 5℄, and [1; 3℄ 6< [3; 5℄. The relation i1 m i2 holds if the right endpoint of i1 equals the left endpoint of i2 and one of those endpoints is open while the other is closed. For example,
[1; 3℄ m (3; 5℄ and [1; 3) m [3; 5℄ but [1; 3℄ m= [3; 5℄ and [1; 3) m= (3; 5℄. The relation i1 o i2 holds if

 either the left endpoint of i1 precedes the left endpoint of i2 or they are equal while the former is closed and the latter is open, either the left endpoint of i2 precedes the right endpoint of i1 or they are equal while both endpoints are closed, and either the right endpoint of i1 precedes the right endpoint of i2 or they are equal while the former is open and the latter is closed.
For example, [1; 3℄ o [2; 4℄, [1; 3℄ o (1; 4℄, [1; 2℄ o [2; 4℄, and [1; 4) o [2; 4℄, but [1; 3℄ o= [1; 4℄,
[1; 2) o= [2; 4℄, and [1; 4℄ o= [2; 4℄. The relation i1 s i2 holds if the left endpoints of i1 and i2 are equal and have the same openness and

 either the right endpoint of i1 precedes the right endpoint of i2 or they are equal while the former is open and the latter is closed.
For example, [1; 3℄ s [1; 4℄, (1; 3℄ s (1; 4℄, and [1; 3) s [1; 3℄, but [1; 3℄ s= (1; 4℄, [1; 3℄ s= [1; 3℄,
[1; 3) s= [1; 3), and [1; 3℄ s= [1; 3). The relation i1 f i2 holds if the right endpoints of i1 and i2 are equal and have the same openness and

 either the left endpoint of i1 follows the left endpoint of i2 or they are equal while the former is open and the latter is closed.
For example, [2; 4℄ f [1; 4℄, [2; 4) f [1; 4), and (2; 4℄ f [2; 4℄, but [2; 4) =f [1; 4℄, (2; 4℄ =f (2; 4℄,
[2; 4℄ =f [2; 4℄, and [2; 4℄ =f (2; 4℄. The relation i1 d i2 holds if

 either the left endpoint of i1 follows the left endpoint of i2 or they are equal while the former is open and the latter is closed and either the right endpoint of i1 precedes the right endpoint of i2 or they are equal while the former is open and the latter is closed.
For example, [2; 3℄ d [1; 4℄ and (1; 4) d [1; 4℄, but [1; 4) d= [1; 4℄, (1; 4℄ d= [1; 4℄, (1; 4) d= (1; 4℄, and
(1; 4) d= [1; 4). The inverse Allen relations >, mi, oi, si, fi, and di are defined analogously to the <,
m, o, s, f, and d relations respectively with the arguments reversed.

The above definitions can be stated more precisely as follows:

1

[q1; r1℄1 =2

[q2; r2℄2

4

$$\left(q_{1}\,=\,q_{2}\right)\wedge\left(\,\alpha_{1}\,=\,\alpha_{2}\,\right)\wedge\left(r_{1}\,=r_{2}\right)\wedge\left(\beta_{1}\,=\,\beta_{2}\,\right)$$
$$\alpha_{1}\left[q_{1},r_{1}\right]_{\beta_{1}}\,<\alpha_{2}\,\left[q_{2},r_{2}\right]_{\beta_{2}}$$

= r1 (:1^:2 ) q2 (2)
$$\begin{array}{r l}{\underline{{{\Delta}}}}&{{}r_{1}\,\leq_{(\,\neg\beta_{1}\wedge\neg\alpha_{2}\,)}\,q_{2}}\end{array}$$
$$(1)^{\frac{1}{2}}$$