The Graph Menagerie: Abstract Algebra and the Mad Veterinarian

• Published in 2011
• Added on 2019-11-06

This article begins with a fanciful concept from recreational mathematics: a machine that can transmogrify a single animal of a given species into a finite nonempty collection of animals from any number of species. Given this premise, a natural question arises: if a Mad Veterinarian has a finite slate of such machines, then which animal menageries are equivalent? To answer this question, the authors associate to the slate of machines a directed "Mad Vet" graph. They then show that the corresponding collection of equivalence classes of animal menageries forms a semigroup and use the structure of the Mad Vet graph to determine when this collection is actually a group. In addition, the authors show that the Mad Vet groups can be identified explicitly using the Smith normal form of a matrix closely related to the incidence matrix of the Mad Vet graph.

Links

• https://www.maa.org/programs/maa-awards/writing-awards/the-graph-menagerie-abstract-algebra-and-the-mad-veterinarian
• https://www.maa.org/sites/default/files/pdf/upload_library/22/Allendoerfer/Abrams2011.pdf

Other information

BibTeX entry

@article{TheGraphMenagerieAbstractAlgebraAndTheMadVeterinarian,
	key = {TheGraphMenagerieAbstractAlgebraAndTheMadVeterinarian},
	type = {article},
	title = {The Graph Menagerie: Abstract Algebra and the Mad Veterinarian},
	author = {Gene Abrams and Jessica K. Sklar},
	abstract = {This article begins with a fanciful concept from recreational mathematics: a machine that can transmogrify a single animal of a given species into a finite nonempty collection of animals from any number of species. Given this premise, a natural question arises: if a Mad Veterinarian has a finite slate of such machines, then which animal menageries are equivalent? To answer this question, the authors associate to the slate of machines a directed "Mad Vet" graph. They then show that the corresponding collection of equivalence classes of animal menageries forms a semigroup and use the structure of the Mad Vet graph to determine when this collection is actually a group. In addition, the authors show that the Mad Vet groups can be identified explicitly using the Smith normal form of a matrix closely related to the incidence matrix of the Mad Vet graph.},
	comment = {},
	date_added = {2019-11-06},
	date_published = {2011-12-07},
	urls = {https://www.maa.org/programs/maa-awards/writing-awards/the-graph-menagerie-abstract-algebra-and-the-mad-veterinarian,https://www.maa.org/sites/default/files/pdf/upload{\_}library/22/Allendoerfer/Abrams2011.pdf},
	collections = {Animals,Attention-grabbing titles,Easily explained,Puzzles},
	url = {https://www.maa.org/programs/maa-awards/writing-awards/the-graph-menagerie-abstract-algebra-and-the-mad-veterinarian https://www.maa.org/sites/default/files/pdf/upload{\_}library/22/Allendoerfer/Abrams2011.pdf},
	urldate = {2019-11-06},
	year = 2011
}