Zero-dimensional MT-algebras: weakening T_{1/2} to T_0

Determine whether the T_{1/2}-algebra hypothesis in the characterization of zero-dimensional MT-algebras can be weakened to a T_0-algebra hypothesis. Specifically, ascertain whether the following implication holds: if an MT-algebra M is T_0 and for every open element a in O(M) we have a = ⋁{ b ∈ CL(M) | b ≤ a } (i.e., a is the join of clopen elements below it), then M is zero-dimensional (equivalently, M is T_1 and every open element is the join of clopen elements below it).

Background

Section 7 introduces zero-dimensional MT-algebras, defined as MT-algebras that are T1 and for which each open element is the join of clopen elements below it. Lemma 7.2 (restrictiontoT1/2) proves an equivalent characterization: M is zero-dimensional if and only if M is a T1/2-algebra and every open element is the join of clopen elements below it.

Immediately following this lemma, the authors pose the question of whether the T1/2 assumption can be weakened to T0 while retaining the same join-of-clopens condition. They note that in the spatial case (atomic MT-algebras corresponding to topological spaces), the weakening to T0 does imply T1, mirroring standard topological results for zero-dimensional T0 spaces.