The intuition of Proposition 4 is that for any fixed observation o, we can extend the observed simult relation with the identity relation to construct the equivalence relation
≃o. The relation ≃o can then be used to partition the set of event occurrences, where we can think of each equivalence class as a "composite event occurrence" consisting of only atomic event occurrences that are pairwise observed *simultaneously* within o.

For example, Fig. 2.3 shows a stratified order ✁o induced by an observation o and the observed before relation. In this case, the equivalence classes of ≃o are the sets
{o1,o2}, {o3}, {o4,o5,o6}, {o7,o8} and {o9,o10}, where the fact that o1 and o2 belong to the same equivalence class means they are observed simultaneously within o.

![0_image_0.png](0_image_0.png)

Fig. 1: A example of a stratified order ✁o induced by an observation o and the observed before relation. (Edges resulted from transitivity are omitted in this diagram for simplicity.)
Proposition 5. If A and B are two distinct equivalence classes of ≃o*, then either A* ×
B ⊆ ✁o *or B*×A ⊆ ✁o. Proof. We pick a ∈ A and b ∈ B. Clearly, a ✁o b or b ✁o a, otherwise a ⌢o b which contradicts that a,b are elements from two distinct equivalence classes. There are two cases:

1. If a ✁o b: we want to show A × B ⊆ ✁o. Let c ∈ A and d ∈ B, it suffices to show c✁o d. Assume for contradiction that ¬(c✁o d). Since c 6≃o d, it follows that d ✁o c.

There are three different subcases:
(a) If a = c, then d ✁o a and a✁o b. Hence, d ✁o b. This contradicts that d,b ∈ B.

(b) If b = d, then b✁o c and a✁o b. Hence, a✁o c. This contradicts that a,c ∈ A.

(c) If a 6= c and b 6= d, then a ⌢o c and b ⌢o d and ¬(a ⌢o d) and ¬(c ⌢o b).

Since ¬(a ⌢o d), either a✁o d or d ✁o a.

- If a✁o d: since d ✁o c, it follows a✁o c. This contradicts a ⌢o c. - If d ✁o a: since a✁o b, it follows d ✁o b. This contradicts d ⌢o b.

Therefore, we conclude A×B ⊆ ✁o.

2. If b✁o a: using a symmetric argument, it follows that B×A ⊆ ✁o. ⊓⊔
Proposition 5 leads to the following consequence. For any observation o, let us define the relation ✁bo on the set Eo = {[a]≃o
: event occurrence(a)} as

$$\mathbf{\hat{l}}_{o}\ {\stackrel{a q}{=}}$$

✁bo df= {(A,B) : A×B ⊆ ✁o ∧ A ∈ Eo ∧ B ∈ Eo}.