Geodesic
Inverse semigroups generated by group congruences. The M\"{o}bius functions
Algebra and discrete mathematics, Tome 16 (2013) no. 1, pp. 116-126

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The computation of the Möbius function of a Möbius category that arises from a combinatorial inverse semigroup has a distinctive feature. This computation is done on the field of finite posets. In the case of two combinatorial inverse semigroups, order isomorphisms between corresponding finite posets reduce the computation to one of the semigroups. Starting with a combinatorial inverse monoid and using a group congruence we construct a combinatorial inverse semigroup such that the Möbius function becomes an invariant to this construction. For illustration, we consider the multiplicative analogue of the bicyclic semigroup and the free monogenic inverse monoid.
Keywords: combinatorial inverse semigroup, Möbius function, Möbius category.
Mots-clés : group congruence 
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E. D. Schwab. Inverse semigroups generated by group congruences. The M\"{o}bius functions. Algebra and discrete mathematics, Tome 16 (2013) no. 1, pp. 116-126. http://geodesic.mathdoc.fr/item/ADM_2013_16_1_a12/