TITLE: Inclusions among diassociativity-related loop properties AUTHOR: Warren D. Smith DATE: 10 MArch 2004 ABSTRACT: 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$\Longrightarrow$2SI'' question of whether a left- and right-alternative loop necessarily has 2-sided inverses. We construct an infinite loop in which this is false. However, $X$ must have a 2-sided inverse in any LRalt loop with $\le 50$ elements or in which $X^n = 1$ with $n \le 13$, results suggesting this is the case in all finite loops. The problem of fully resolving this may be the hardest natural problem in mathematics that is this simply posed.