A1 Refereed original research article in a scientific journal
Defect theorems with compatibility relations
Authors: Halava V, Harju T, Kärki T
Publisher: SPRINGER
Publication year: 2008
Journal: Semigroup Forum
Journal name in source: SEMIGROUP FORUM
Journal acronym: SEMIGROUP FORUM
Volume: 76
Issue: 1
First page : 1
Last page: 24
Number of pages: 24
ISSN: 0037-1912
DOI: https://doi.org/10.1007/s00233-007-9013-9
Abstract
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.
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.
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