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

: 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.