<p>This paper will use the kernel-trace approach of congruences to investigate one-sided congruences on a class of inverse semigroups. The gauge inverse submonoids, as kernels of congruences, play a leading role in this investigation. A congruence whose kernel is the gauge inverse submonoid and whose trace is the universal relation is a group congruence. The quotient group is isomorphic to the additive group of integers. This group congruence generates a Möbius category such that the corresponding breaking process preserves the Möbius function.</p>