Discussion. LG is a weaker logic than CNL, the Classical Non-Associative Lambek Calculus of de Groote and Lamarche 2002. In the latter system, like in classical linear logic, we have an involutive negation and De Morgan dualities turning multiplicative conjunction (times) and disjunction (par) into interexpressible operations. With respect to the linguistic applications, one can compare LG with the Discontinuous Calculus of §3.2.  Whereas the latter provides uniform analyses of both the extraction and the infixation type of discontinuities, the distributivity principles of LG are primarily addressing the infixation variety.  It may be interesting in this respect to note that Grishin 1983 proposes a second group of distributivity principles — converses of the ones discussed above.  Whether these could play a role in the analysis of overt displacement remains to be seen.  From a formal point of view, each of the groups of distributivities is a conservative extension of the basic symmetric calculus.  But the combination of the two (i.e., the distributivity principles as invertible rules) induces partial associativity/commutativity of the (co)product operations, i.e., structure-preservation is lost.