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.
📌 3 Key Takeaways
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Objective & Hypothesis:Every gauge inverse submonoid (including Jones-Lawson's gauge inverse submonoid of the polycyclic monoid $P_{n}$) is a normal submonoid.
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Key Findings & Data:In 2018, Alyamani and Gilbert introduced an equivalence relation on an inverse semigroup associated to a normal inverse subsemigroup.
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Conclusion & Impact: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.
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.
Schwab, E. D. (2022). The bicyclic semigroup as the quotient inverse semigroup by any gauge inverse submonoid. Journal of Algebra Combinatorics Discrete Structures and Applications, Cilt: 9 Sayı: 2, 53–61. https://doi.org/10.13069/jacodesmath.1112177