  We build on the description of left congruences on an inverse semigroup in
terms of the kernel and trace due to Petrich and Rankin. The notion of an
inverse kernel for a left congruence is developed. Various properties of both
the trace and inverse-kernel are discussed, in particular that both the trace
and inverse-kernel maps are onto $\cap$-homomorphisms. The lattice of left
congruences is identified as a subset of the direct product of the lattice of
congruences on the idempotents and the lattice of full inverse subsemigroups.
We use this to describe the lattice of left congruences on the bicyclic monoid.
  It is shown that that every finitely generated left congruence is the join of
a finitely generated trace minimal left congruence and a finitely generated
idempotent separating left congruence. Characterisations of inverse semigroups
that are left Noetherian, or such that the universal congruence is finitely
generated, are given.
