Defect theorems with compatibility relations

: Halava V, Harju T, Kärki T

Publisher: SPRINGER

: 2008

: Semigroup Forum

: SEMIGROUP FORUM

: 76

: 1

: 24

: 0037-1912

DOI: https://doi.org/10.1007/s00233-007-9013-9

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 Schutzenberger 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.

Mathematics