The lattice of one-sided congruences on an inverse semigroup

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 the trace and inverse kernel are discussed, in particular that the inverse kernel is a full inverse subsemigroup and that both the trace and inverse kernel maps are onto ∩ -homomorphisms. It is shown that a left congruence is determined by its trace and inverse kernel, and 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 demonstrate 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 are given of inverse semigroups that are left Noetherian, or are such that Rees left congruences are finitely generated.