First day of Mile High

Today was the first day of the Third Mile High Conference on Nonassociative Mathematics at University of Denver. Looks like a good group of people, only a few of whom I know. Yay for making connections.

Had lunch with Aleš Drápal today; I don't believe that we'd ever actually met before, though he immediately knew who I was, as I expected. The world is still small.

Neither he nor I are actively working on LD-related research at the moment, though finite LDs form nice test examples for computational packages; he mentioned that he has some unpublished theorems relating to the "levels" of homomorphisms between Laver tables. He also mentioned a direction of possible research that he had mostly abandoned, but where there might be results of manageable difficulty for a person or collaboration with the right crossover knowledge:

Let \( \mathbf{A} = \langle A; * \) be a LD-groupoid (probably finite and monogenerated). It is known that in some cases, such as when \( \mathbf{A} \) is a Laver Table, we can define an associative composition \( \circ \) on \( A \) satisfying

• \( ( a \circ b ) * c = a * (b * c) \)
• \( a \circ b = (a * b) \circ a \)
• \( a * (b \circ c) = ( a * b ) \circ (a * c) \)

and that this operation is provably unique in the case of Laver Tables. (If \( \mathbf{A} \) is a Laver Table, then \( (a \circ b)^+ = a * (b^+) \).) Aleš thinks that the semigroups obtainable in this way may be interesting from a semigroup theory perspective, but couldn't find any collaborators in the 90s whom he could interest in looking at the problem in any level of detail. (He thinks there's less there in terms of the universal algebraic properties of these semigroups -- what kind of varieties or quasivarieties they generate, blah blah blah...)

To future me, who wants to play around with this problem: start with Aleš' paper in Semigroup Forum 51, "On the semigroup structure of cyclic left distributive algebras".