Tools and Techniques in Modal Logic Marcus Kracht II ...

68 2. Fundamentals of Modal Logic I
that is, that it commutes with the modal operators. Show that it is sufficient.
Exercise 43. Show that if in Definition 2.4.7 we did not require π −1 to be surjective,
then there were injective p–morphisms which are not embeddings.
Exercise 44. Let π : F ↠ G be a p–morphism and x ∈ g. Show that if there
is no chain of length k x = x0 ⊳ j x1 ⊳ j x2 . . . ⊳ j xk then the same holds for π(x)
as well. Show that if π(x) ⋪ j π(x) then π −1 (π(x)) is a j–antichain, that is, for all
y, z ∈ π −1 (π(x)) y ⋪ j z.
Exercise 45. Prove Proposition 2.4.5 and Theorem 2.4.6.
Exercise 46. Formulate and prove Proposition 2.4.5 and Theorem 2.4.6 for (generalized)
frames instead of Kripke–frames.
Exercise 47. Let f = 〈 f, 〈⊳ j : j < κ〉〉 be a Kripke–frame and G a subgroup of the
group Aut(f) of automorphisms of f. Put
[x] := {y : there exists g ∈ G : g(x) = y} .
Also, put [x] ⊳ j [y] if there exist �x ∈ [x] and �y ∈ [y] such that �x ⊳ j �y. Show that
x ↦→ [x] is a p–morphism. (An automorphism of f is a bijective p–morphism from f
to f. The automorphisms of a structure generally form a group.)
2.5. Some Important Modal Logics
Among the infinitely many logics that can be considered there are a number of
logics that are of fundamental importance. Their importance is not only historical
but has as we will see also intrinsic reasons. We begin with logics of a single operator.
Here is a list of axioms together with their standard names. (In some cases we
have given alternate forms of the axioms. The first is the one standardly known, the
second a somewhat more user friendly variant.)