Defect theorems with compatibility re- lations

We consider words together with a compatibility relation induced by a relation on letters. Unique factorization with respect to two arbitrary word relations R and S defines the (R, S)-freeness of the semigroup considered. We generalize the stability theorem of Schützenberger and Tilson’s closure result for (R, S)free semigroups. The inner and the outer (R, S)-unique factorization hull and the (R, S)-free hull of a set of words are introduced and we show how they can be computed. We prove that the (R, S)-unique factorization hulls possess a defect effect, which implies a variant of a cumulative defect theorem of word semigroups. In addition, a defect theorem of partial words is proved as a corollary.