  In recent years, the compositional distributional approach in computational
linguistics has opened the way for an integration of the \emph{lexical} aspects
of meaning into Lambek's type-logical grammar program. This approach is based
on the observation that a sound semantics for the associative, commutative and
unital Lambek calculus can be based on vector spaces by interpreting fusion as
the tensor product of vector spaces.
  In this paper, we build on this observation and extend it to a `vector space
semantics' for the \emph{general} Lambek calculus, based on \emph{algebras over
a field} $\mathbb{K}$ (or $\mathbb{K}$-algebras), i.e. vector spaces endowed
with a bilinear binary product. Such structures are well known in algebraic
geometry and algebraic topology, since they are important instances of Lie
algebras and Hopf algebras. Applying results and insights from duality and
representation theory for the algebraic semantics of nonclassical logics, we
regard $\mathbb{K}$-algebras as `Kripke frames' the complex algebras of which
are complete residuated lattices.
  This perspective makes it possible to establish a systematic connection
between vector space semantics and the standard Routley-Meyer semantics of
(modal) substructural logics.
