  It was proved by Maksimova in 1977 that exactly eight varieties of Heyting
algebras have the amalgamation property, and hence exactly eight axiomatic
extensions of intuitionistic propositional logic have the deductive
interpolation property. The prevalence of the deductive interpolation property
for axiomatic extensions of substructural logics and the amalgamation property
for varieties of pointed residuated lattices, their equivalent algebraic
semantics, is far less well understood, however. Taking as our starting point a
formulation of intuitionistic propositional logic as the full Lambek calculus
with exchange, weakening, and contraction, we investigate the role of the
exchange rule--algebraically, the commutativity law--in determining the scope
of these properties. First, we show that there are continuum-many varieties of
idempotent semilinear residuated lattices that have the amalgamation property
and contain non-commutative members, and hence continuum-many axiomatic
extensions of the corresponding logic that have the deductive interpolation
property in which exchange is not derivable. We then show that, in contrast,
exactly sixty varieties of commutative idempotent semilinear residuated
lattices have the amalgamation property, and hence exactly sixty axiomatic
extensions of the corresponding logic with exchange have the deductive
interpolation property. From this latter result, it follows also that there are
exactly sixty varieties of commutative idempotent semilinear residuated
lattices whose first-order theories have a model completion.
