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timeabdceventsFigure 4: Cones of informationsSince we have a complete sup semilattice with a bottom element ε, we can geta lattice by the construction recalled in Section 2.2. Applying this construction,it is realized that the operation ∧ graphically amounts to the intersection ofcorresponding areas. For example, a ∩ b = d in Figure 4. It is then obvious thatthis lattice is distributive.Remark 8 By considering that a piece of information is equivalent to the collectionof informations lying in the south-east cone of which that piece is thevertex, we have been able to resolve potential contradictions that would otherwisehave arisen when having to consider b ∪ c for example. But one mayargue that this is not the only may to avoid these contradictions and that settingb ∪ c = c rather than b would have done the same job. This correspondsto saying that a piece of information dominates all informations in the northwest,rather than south-east cone. Otherwise stated, this amounts to turningexpressions as at most (resp. at the earliest) intoat least (resp. at the latest)in sentences above. Indeed, this other way of performing “addition” of informationswill prove to be useful in another context later on. The sum we havechosen is in fact compatible with the way join transitions gather informationscoming from upstream in Petri nets (it is worth taking a while thinking of that).□8.2 Shift operatorsAs it was apparent from earlier sections, the graph carries informations fromupstream to downstream transitions. More precisely, in Figure 1, informations(n, t) x1 can be related to informations (n ′ ,t ′ ) u1 if the latter are shifted backwardsby three units in time, and to informations (n ′′ ,t ′′ ) x2 if these are shiftedbackwards by one unit in the number of events.30