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

Year 2022, Volume: 9 Issue: 2, 53 - 61, 13.05.2022

Emil Daniel Schwab

https://doi.org/10.13069/jacodesmath.1112177

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

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