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Compatibility relations on codes and free monoids

Abstract
A compatibility relation on letters induces a reflexive and symmetric relation on words of equal length. We consider these word relations with respect to the theory of variable length codes and free monoids. We define an (R,S)-code and an (R,S)-free monoid for arbitrary word relations R and S. Modified Sardinas-Patterson algorithm is presented for testing whether finite sets of words are (R,S)-codes. Coding capabilities of relational codes are measured algorithmically by finding minimal and maximal relations. We generalize the stability criterion of Schützenberger and Tilson's closure result for (R,S)-free monoids. The (R,S)-free hull of a set of words is introduced and we show how it can be computed. We prove a defect theorem for (R,S)-free hulls. In addition, a defect theorem of partial words is proved as a corollary.