Groebner Finite Path Algebras

Gr¨obner Finite Path Algebras Micah J. Leamer Thesis submitted to the faculty of Virginia Polytechnic Institute and State University in partial ful£llment of the requirements for the degree of Master of Science in Mathematics Edward Green, Chair Charles Parry John Rossi July 1, 2004 Blacksburg, Virginia Keywords: Groebner Bases, Path Algebra Copyright 2004, Micah J. Leamer Gr¨obner Finite Path Algebras Micah J. Leamer Abstract Let K be a £eld and Γ a £nite directed multi-graph. In this paper I classify all path algebras KΓ and admissible orders with the property that all of their £nitely generated ideals have £nite Gr o¨ bner bases. Acknowledgements First of all I would like to thank Charles Parry and John Rossi for joining my committee on such short notice. They exemplify the supportive environment that the VA Tech math department fosters. Joseph Ball offered up his time and his expertise to meet with me for several weeks and teach me the nuances of operator algebras. He was a faithful committee member until the rescheduling of my defense con¤icted with his prior commitments. Adrian Keister offered me help with many latex problems. Hannah Swiger has been invaluable. She has gone above and beyond the line of duty to help me process forms and meet all of the administrative requirements. She has been positive, reassuring and at all times candid. I would especially like to thank Dean Reiss for encouraging me to join the graduate school at Tech. He also opened doors so that my application process was smooth and simple. I would not have spent this year at Tech nor written this thesis otherwise. Edward Green the chair of my committee has been a core £gure in my mathematical development, as well as the central resource for this thesis. He has considered me valuable enough to meet with on a weekly basis for the past two years. All of our meetings for the past year have been for the purpo