What is the upper part of the lattice of bimodal logics? (Q1327483)

The paper proves a rather surprising fact for the lattice of normal bimodal logics. Namely, let \(\Lambda_ 1\) be the monomodal logic of a two-element chain, and consider the ``fusion'' \(\Lambda_ 1 \otimes \text{S5}\), which is defined as the extension of the minimal normal bimodal logic by axioms of \(\Lambda_ 1\) for \(\square_ 1\) and axioms of S5 for \(\square_ 2\). For a modal logic \(L\), \({\mathcal E} (L)\) denotes the lattice of normal extensions of \(L\) (in the same language). The main theorem states that \({\mathcal E} (T)\) (\(T\) is the logic of all reflexive Kripke frames) can be embedded into \({\mathcal E} (\Lambda_ 1 \otimes \text{S5}\)). Thus the latter is uncountable, although \({\mathcal E} (\Lambda_ 1)\) is finite and \({\mathcal E} (S5)\) is a \((1+ \omega^*)\)-chain.