Relativized Adjacency

For each class in the piecewise-local subregular hierarchy, a relativized (tier-based) variant is defined. Algebraic as well as automata-, language-, and model-theoretic characterizations are provided for each of these relativized classes, except in cases where this is provably impossible. These various characterizations are necessarily intertwined due to the well-studied logic-automaton connection and the relationship between finite-state automata and (syntactic) semigroups. Closure properties of each class are demonstrated by using automata-theoretic methods to provide constructive proofs for the closures that do hold and giving language-theoretic counterexamples for those that do not. The net result of all of this is that, rather than merely existing as an operationally-defined parallel set of classes, these relativized variants integrate cleanly with the other members of the piecewise-local subregular hierarchy from every perspective. Relativization may even prove useful in the characterization of star-free, as every star-free stringset is the preprojection of another (also star-free) stringset whose syntactic semigroup is not a monoid.