Necessity of monotonicity for the lattice property of the sum

Determine whether the proof that the sum of a modular connected system is a lattice depends on the monotonicity condition (MC8).

Background

The paper proves that the sum of a modular connected system is a lattice, with meet and join operations constructed from the block structure and the skeleton. The proof of Theorem 4.66 proceeds through Lemmas 4.60 and 4.58, whose arguments rely on the finiteness of certain intervals established in Lemma 4.40; that finiteness ultimately depends on the monotonicity condition (MC8).

The author explicitly suggests that monotonicity may not be intrinsically necessary for the lattice conclusion, while acknowledging that the proof as given depends on it. The unresolved problem is therefore to determine whether the conclusion remains valid for connected systems lacking (MC8), or whether a counterexample exists.