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

Abstract

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.

Keywords

• Inverse semigroup
• Ordered groupoid
• Gauge inverse submonoid
• Bicyclic semigroup

References

1. [1] N. Alyamani, N. D. Gilbert, Ordered groupoid quotients and congruences on inverse semigroups, Semigroup Forum 96 (2018) 506–522.
2. [2] D. G. Jones, M. V. Lawson, Strong representations of the polycyclic inverse monoids: Cycles and atoms, Period. Math. Hung. 64 (2012) 54–87.
3. [3] M. V. Lawson, Inverse semigroups: the theory of partial symmetries, World Scientific, Singapore (1998).
4. [4] E. D. Schwab, Möbius monoids and their connection to inverse monoids, Semigroup Forum 90 (2015) 694–720.
5. [5] E. D. Schwab, Gauge inverse monoids, Algebra Colloq. 27(2) (2020) 181–192.