Open AccessWalker's Cancellation Theorem
Author(s) -
Robert S. Lubarsky,
Fred Richman
Publication year - 2018
Publication title -
epic series in computing
Language(s) - English
Resource type - Conference proceedings
ISSN - 2398-7340
DOI - 10.29007/vz4n
Subject(s) - abelian group , mathematics , constructive , constructive proof , endomorphism , discrete mathematics , combinatorics , pure mathematics , computer science , process (computing) , operating system
Walker's cancellation theorem says that if B + Z is isomorphic to C + Z in the category of abelian groups, then B is isomorphic to C. We construct an example in a diagram category of abelian groups where the theorem fails. As a consequence, the original theorem does not have a constructive proof. In fact, in our example B and C are subgroups of Z<sup>2</sup>. Both of these results contrast with a group whose endomorphism ring has stable range one, which allows a constructive proof of cancellation and also a proof in any diagram category.