  We extend the theory of unified correspondence to a very broad class of
logics with algebraic semantics given by varieties of normal lattice expansions
(LEs), also known as `lattices with operators'. Specifically, we introduce a
very general syntactic definition of the class of Sahlqvist formulas and
inequalities, which applies uniformly to each LE-signature and is given purely
in terms of the order-theoretic properties of the algebraic interpretations of
the logical connectives. Together with this, we introduce a variant of the
algorithm ALBA, specific to the setting of LEs, which effectively computes
first-order correspondents of LE-inequalities, and is guaranteed to succeed on
a wide class of inequalities (the so-called inductive inequalities) which
significantly extend the Sahlqvist class. Further, we show that every
inequality on which ALBA succeeds is canonical. The projection of these results
yields state-of-the-art correspondence theory for many well known substructural
logics, such as the Lambek calculus and its extensions, the Lambek-Grishin
calculus, the logic of (not necessarily distributive) de Morgan lattices, and
the multiplicative-additive fragment of linear logic.

Logic