Pattern Structures and Their Projections

Pattern structures consist of objects with descriptions (called patterns) that allow a semilattice operation on them. Pattern structures arise naturally from ordered data, e.g., from labeled graphs ordered by graph morphisms. It is shown that pattern structures can be reduced to formal contexts, however sometimes processing the former is often more efficient and obvious than processing the latter. Concepts, implications, plausible hypotheses, and classifications are defined for data given by pattern structures. Since computation in pattern structures may be intractable, approximations of patterns by means of projections are introduced. It is shown how concepts, implications, hypotheses, and classifications in projected pattern structures are related to those in original ones.