  We investigate language interpretations of two extensions of the Lambek
calculus: with additive conjunction and disjunction and with additive
conjunction and the unit constant. For extensions with additive connectives, we
show that conjunction and disjunction behave differently. Adding both of them
leads to incompleteness due to the distributivity law. We show that with
conjunction only no issues with distributivity arise. In contrast, there exists
a corollary of the distributivity law in the language with disjunction only
which is not derivable in the non-distributive system. Moreover, this
difference keeps valid for systems with permutation and/or weakening structural
rules, that is, intuitionistic linear and affine logics and affine
multiplicative-additive Lambek calculus. For the extension of the Lambek with
the unit constant, we present a calculus which reflects natural algebraic
properties of the empty word. We do not claim completeness for this calculus,
but we prove undecidability for the whole range of systems extending this
minimal calculus and sound w.r.t. language models. As a corollary, we show that
in the language with the unit there exissts a sequent that is true if all
variables are interpreted by regular language, but not true in language models
in general.
