Self-Distributive Magmas and Their Richter's Degrees

Richter's degree of a countable algebraic model is the Turing degree theoretic measure of the complexity of its isomorphism class. It is defined to be the least Turing degree in the set of Turing degrees of all isomorphic copies of the model. It has been shown that some models, such as abelian groups or partially ordered sets, have arbitrary Richter's degrees, by showing that their isomorphism classes contain infinite anti-chains of models with certain algebraic and computability theoretic properties. Some theories have a model with its isomorphism class containing a minimal pair, which means that the isomorphism class of the model does not have the least element or Richter's degree. For models that do not have Richter's degrees, their Turing jump degrees have been investigated. I extend these results to models not previously studied in computability theory, such as self-distributative magmas of interest in low-dimensional topology. These self-distributive magmas do not have to be associative. In particular, I investigate racks and crossed sets and show that these classes contain models with arbitrary Richter's degrees, as well as models with no Richter's degree.