# 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 μ*A* 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 .

**Lemma**

For any formula *A*, and .

**Lemma** (Functoriality)

If *A* and *B* are two formulas such that then, for any exponential modality μ, .

**Lemma**

For any formula *A*, and .

**Lemma**

For any formula *A*, 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.

*Proof.*
We obtain by functoriality from (and similarly for ).
From , we deduce by functoriality and (with ). In a similar way we have and .

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.

*Proof.*
We have already seen and . By functoriality we deduce and by we deduce .

The others are obtained from these ones by duality: entails .

**Lemma**

If α is an atom, and
.

We can prove that no other relation between classes is true (by relying on the previous lemma).

*Proof.*
From the lemma and , we have .

Then cannot be smaller than any other of the seven modalities (since they are all smaller than or ). For the same reason, cannot be smaller than , , or . This entails that is only smaller than since it is not smaller than (by duality from not smaller than ).

From these, and , we deduce that no other relation is possible.

The order relation on equivalence classes of exponential modalities is a lattice.