  In this paper we extend the research programme in algebraic proof theory from
axiomatic extensions of the full Lambek calculus to logics algebraically
captured by certain varieties of normal lattice expansions (normal LE-logics).
Specifically, we generalise the residuated frames in [34] to arbitrary
signatures of normal lattice expansions (LE). Such a generalization provides a
valuable tool for proving important properties of LE-logics in full uniformity.
We prove semantic cut elimination for the display calculi D.LE associated with
the basic normal LE-logics and their axiomatic extensions with analytic
inductive axioms. We also prove the finite model property (FMP) for each such
calculus D.LE, as well as for its extensions with analytic structural rules
satisfying certain additional properties.
