For any regular semigroup S
the relation 𝜃 is defined on the lattice, Λ(S), of congruences on S by: (ρ,τ) ∈ 𝜃 if and
only if ρ and τ induce the same partition of the idempotents of S. Then 𝜃 is an
equivalence relation on Λ(S) such that each equivalence class is a complete modular
sublattice of Λ(S). If S is an inverse semigroup then 𝜃 is a congruence on Λ(S),
Λ(S)∕𝜃 is complete and the natural homomorphism of Λ(S) onto Λ(S)∕𝜃 is a
complete lattice homomorphism. Any congruence on an inverse semigroup S can be
characterized in terms of its kernel, namely, the set of congruence classes containing
the idempotents of S. In particular, any congruence on S induces a partition of the
set ES of idempotents of S satisfying certain normality conditions. In this note, those
partitions of ES which are induced by congruences on S and the largest and
smallest congruences on S correspond ing so such a partition of ES are
characterized.
