Locally tabular products of modal logics

In the product L₁× L₂ of two Kripke complete consistent logics, local tabularity of L₁ and L₂ is necessary for local tabularity of L₁× L₂. However, it is not sufficient: the product of two locally tabular logics can be not locally tabular. We provide extra semantic and axiomatic conditions which give criteria of local tabularity of the product of two locally tabular logics. Then we apply them to identify new families of locally tabular products.