We present some homological properties of a relation β on ordered groupoids that generalises the minimum group congruence for inverse semigroups. When β is a transitive relation on an ordered groupoid G, the quotient G / β is again an ordered groupoid, and construct a pair of adjoint functors between the module categories of G and of G / β. As a consequence, we show that the homology of G is completely determined by that of G / β, generalising a result of Loganathan for inverse semigroups.
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Bainson et al. (2020) studied this question.
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