The realization of refinement monoids

• Thesis title in Czech: Realizace zjemňujících monoidů
• Thesis title in English: The realization of refinement monoids
• Key words: Zjemňující monoid|Von Neumannovsky regulární okruh|Separovaný graf|Problém realizace|Leavittova algebra cest
• English key words: Refinement monoid|Von Neumann regular ring|Separated graph|Realization problem|Leavitt path algenbra
• Academic year of topic announcement: 2023/2024
• Thesis type: dissertation
• Thesis language: angličtina
• Department: Department of Algebra (32-KA)
• Supervisor: doc. Mgr. Pavel Růžička, Ph.D.
• Author: hidden - assigned and confirmed by the Study Dept.
• Date of registration: 18.09.2023
• Date of assignment: 18.09.2023
• Confirmed by Study dept. on: 18.09.2023

Guidelines

Recently Ara, Bosa, and Pardo [3] proved that every finitely generated conical refinement monoid is representable as the monoid of isomorphism classes of finitely generated projective modules over a von Neumann regular ring. On the other hand F. Wehrung [5] constructed an example of a non-representable conical refinement monoid of size aleph 2. Whether every countable conical refinement monoid is representable remains open. Solving this would contribute to solutions of several long standing problems (see [2]). The student should get familliar with the topic, related arreas, and possibly attack the countable case of the representation problem.

References

1. G Abrams, P Ara, MS Molina, Leavitt path algebras, Springer, 2017
2. P. Ara, The realization problem for von Neumann regular rings, Ring Theory 2007, (2008), 21-37
3. P. Ara, J. Bosa, E. Pardo, The realization problem for finitely generated refinement monoids, Selecta Mathematica vol. 26, 33 (2020) ,
4. K. R. Goodearl, Von Neumann regular rings, Pitman, 1979
5. F. Wehrung, Non-measurability properties of interpolation vector spaces, Israel Journal of Mathematics 103, (1998), 177-206.