Acyclic modal logic: an exercise in irreflexivization

Given a unimodal logic Λ extending the modal logic of reflexive frames T = K⊕2p→ p (the symbol ⊕ is used in the same way as in [3]), we can define several irreflexive or non-reflexive counterparts of Λ. Our first approach will be semantical. As usual, given a set W , define ∆W := {〈x, x〉 | x ∈W}. Definition 1.1 The irreflexivization of a frame F = 〈W,R〉 is 〈W,R〉• := 〈W,R − ∆W 〉. For X a class of frames, X• := {F• | F ∈ X}. The irreflexive counterpart of Λ, Λ• := Log(Mod•(Λ)) is the logic determined by irreflexivizations of frames for Λ.