Bimodal logics with a 'weakly connected' component without the finite model property

There are two known general results on the finite model property (fmp) of commutators [L0, L1] (bimodal logics with commuting and confluent modalities). If L is finitely axiomatisable by modal formulas having universal Horn first-order correspondents, then both [L,K] and [L,S5] are determined by classes of frames that admit filtration, and so have the fmp. On the negative side, if both L0 and L1 are determined by transitive frames and have frames of arbitrarily large depth, then [L0, L1] does not have the fmp. In this paper we show that commutators with a ‘weakly connected’ component often lack the fmp. Our results imply that the above positive result does not generalise to universally axiomatisable component logics, and even commutators without ‘transitive’ components such as [K3,K] can lack the fmp. We also generalise the above negative result to cases where one of the component logics has frames of depth one only, such as [S4.3,S5] and the decidable product logic S4.3×S5. We also show cases when already half of commutativity is enough to force infinite frames.