The lattice of one-sided congruences on an inverse semigroup (Q6057462)

It is well known that congruences and left congruences on an inverse semigroup \(S\) may be described in terms of their traces (restriction to the semilattice \(E\) of idempotents) and kernels (union of the congruence classes containing idempotents). In the present paper, the inverse kernel \(\mathrm{Iker}\rho=\{a\mid a\#{\rho}aa^{-1}\}\) of a left congruence \(\rho\) on \(S\) is defined. It is an inverse subsemigroup of the kernel and its construction is shown to have nice properties, such as preserving arbitrary intersections (which the kernel construction does not) and, together with the trace, still completely determining the left congruence. The author discusses extensively how this relates with the classical kernel trace approach. Here is a summary of further results derived from the new approach: a characterization of the lattice of left congruences on \(S\) with a given trace; a characterization of the traces of the left congruences on \(S\) whose inverse kernel is a given full inverse subsemigroup of \(S\); a description of the lattice of left congruences on \(S\) in terms of the inverse kernel trace approach and what corresponds to finitely generated left congruences; a characterization of when the Rees left congruence determined by a left ideal \(A\) is finitely generated.