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 [L,L'] (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 L and L' are determined by transitive frames and have frames of arbitrarily large depth, then [L,L'] 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 [K.3,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 S4.3xS5. We also show cases when already half of commutativity is enough to force infinite frames.