Some classes of sets of structures definable without quantifiers

We derive abstract characterizations of the Strictly Piecewise Local (SPL) and Piecewise Locally Testable (PLT) stringsets. These generalize both the Strictly Local/Locally Testable stringsets (SL and LT) and Strictly Piecewise/Piecewise Testable stringsets (SP and PT) in that SPL constraints can be stated in terms of both adjacency and precedence. We do this in a fully abstract setting which applies to any class of purely relational models that label the points in their domain with some finite labeling alphabet. This includes, for example, labeled trees and graphs. The actual structure of the class of intended models only shows up in interpreting the abstract characterizations of the definable sets in terms of the structure of the models themselves.