### In book

We consider propositional normal unimodal *pretransitive* logics, i.e., logics with expressible `transitive' modality. There is a long-standing open problem about the finite model property (fmp) and decidability of pretransitive logics, in particular - for the logics **K**^*m*_*n *= **K**+[]^*m* *p* -> []^n p, *n*>*m*>1.

In this paper we introduce public announcements to Subset Space Logic (SSL). In order to do this we have to change the original semantics for SSL a little and consider a weaker version of SSL without the cross axiom. We present an axiomatization, prove completeness and show that this logic is PSPACE-complete. Finally, we add the arbitrary announcement modality which expresses ``true after any announcement'', prove several semantic results, and show completeness for a Hilbert-style axiomatization of this logic.

We consider modal logics of products of neighborhood frames and prove that for any pair L and L' of logics from set {S4, D4, D, T} modal logic of products of L-neighborhood frames and L'-neighborhood frames is the fusion of L and L'.