The bicyclic semigroup as the quotient inverse semigroup by any gauge inverse submonoid

  • Emil Daniel Schwab
Keywords: Inverse semigroup, Ordered groupoid, Gauge inverse submonoid, Bicyclic semigroup


Every gauge inverse submonoid (including Jones-Lawson's gauge inverse submonoid of the polycyclic monoid $P_{n}$) is a normal submonoid. In 2018, Alyamani and Gilbert introduced an equivalence relation on an inverse semigroup associated to a normal inverse subsemigroup. The corresponding quotient set leads to an ordered groupoid. In this note we shall show that this ordered groupoid is inductive if the normal inverse subsemigroup is a gauge inverse submonoid and the corresponding quotient inverse semigroup by any guage inverse submonoid is isomorphic either to the bicyclic semigroup or to the bicyclic semigroup with adjoined zero.


N. Alyamani, N. D. Gilbert, Ordered groupoid quotients and congruences on inverse semigroups, Semigroup Forum 96 (2018) 506–522.

D. G. Jones, M. V. Lawson, Strong representations of the polycyclic inverse monoids: Cycles and atoms, Period. Math. Hung. 64 (2012) 54–87.

M. V. Lawson, Inverse semigroups: the theory of partial symmetries, World Scientific, Singapore (1998).

E. D. Schwab, Möbius monoids and their connection to inverse monoids, Semigroup Forum 90 (2015) 694–720.

E. D. Schwab, Gauge inverse monoids, Algebra Colloq. 27(2) (2020) 181–192.