Lattice of exponential modalities
An exponential modality is an arbitrary (possibly empty) sequence of the two exponential connectives and . It can be considered itself as a unary connective. This leads to the notation μA for applying an exponential modality μ to a formula A.
There is a preorder relation on exponential modalities defined by if and only if for any formula A we have . It induces an equivalence relation on exponential modalities by μ˜ν if and only if and .
This allows to prove that any exponential modality is equivalent to one of the following seven modalities: (the empty modality), , , , , or . Indeed any sequence of consecutive or in a modality can be simplified into only one occurrence, and then any alternating sequence of length at least four can be simplified into a smaller one.
The order relation induced on equivalence classes of exponential modalities with respect to ˜ can be proved to be the one represented on the picture in the top of this page. All the represented relations are valid.
We can prove that no other relation between classes is true (by relying on the previous lemma).
The order relation on equivalence classes of exponential modalities is a lattice.