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In this paper we provide a simplified, possibilistic semantics for the logics K45(G), i.e. a many-valued counterpart of the classical modal logic K45 over the [0, 1]-valued Gödel fuzzy logic $$\mathbf{G}$$ G . More precisely, we characterize K45(G) as the set of valid formulae of the class of possibilistic Gödel frames $$\langle W, \pi \rangle $$ ⟨ W , π ⟩ , where W is a non-empty set of worlds and $$\pi : W \mathop {\rightarrow }[0,1]$$ π : W → [ 0 , 1 ] is a possibility distribution on W. We provide decidability results as well. Moreover, we show that all the results also apply to the extension of K45(G) with the axiom (D), provided that we restrict ourselves to normalised Gödel Kripke frames, i.e. frames $$\langle W, \pi \rangle $$ ⟨ W , π ⟩ where $$\pi $$ π satisfies the normalisation condition $$\sup _{w \in W} \pi (w) = 1$$ sup w ∈ W π ( w ) = 1 .

We characterize commutative idempotent involutive residuated lattices as disjoint unions of Boolean algebras arranged over a distributive lattice. We use this description to introduce a new construction, called gluing, that allows us to build new members of this variety from other ones. In particular, all finite members can be constructed in this way from Boolean algebras. Finally, we apply our construction to prove that the fusion reduct of any finite member is a distributive semilattice, and to show that this variety is not locally finite.