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Algebra |
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Authors: Warren D. Smith
We attempt to find all implications among 19 commonly used diassociativity, Moufang, Bol, alternativity and inverse-related properties in loops. There are 6 among these that appear to be valid in finite but not infinite loops. Under that assumption, we completely settle the problem. We study in detail the apparently-simplest among the 6 nasty cases: the "LRalt⇒2SI" question of whether a left- and right-alternative loop necessarily has 2-sided inverses. We construct an infinite loopin which this is false. However, X must have a 2-sided inverse in any LRalt loop with ≤50 elements or in which Xn=1 with n≤13 – results suggesting this is the case in all finite loops. The problem of fully resolving this might be the hardest natural problem in mathematics that is this simply posed!
Comments: 15 Pages. On www since 2004, now uploading to VIXRA for archival purposes.
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