I'm trying to solve a question which asks me to show that for any two finite strings O₁ and O₂ of □s and ◊s, (e.g. □□◊□◊□), that
i) if O₁≡O₂ then OO₁≡OO₂
ii) if O₁≡O₂ then O₁O≡O₂O
where O is any other string and we say O₁≡O₂ (they express the same modality (under some modal logic system) if ⊨(O₁ϕ↔O₂ϕ).
I've managed to solve the first part quite easily using the fact that if ⊨ϕ₁↔ϕ₂ then ⊨χ(ϕ₁)↔χ(ϕ₂) where χ(ϕ) is just the result of replacing P with ϕ in the formula χ(P).
I'm not sure how to go about solving the second though, because in this case the new string is inserted between the old one and the sentence ϕ, and I can't see an obvious way to do that with this formula.
I'm also trying to find a way to show, using these, that under the modal system B (where the accessibility relation is reflexive and symmetric), that there are infinitely many modalities, using some sort of informal semantic argument.
I've done the same for S4 and S5, but those have finite numbers of modalities so the strategy was different.
I'd really appreciate any help you could offer!