  In a previous work we introduced a non-associative non-commutative logic
extended by multimodalities, called subexponentials, licensing local
application of structural rules. Here, we further explore this system,
exhibiting a classical one-sided multi-succedent classical analogue of our
intuitionistic system, following the exponential-free calculi of Buszkowski,
and de Groote, Lamarche. A large fragment of the intuitionistic calculus is
shown to embed faithfully into the classical fragment.
