Complementation of Rational Sets on Countable Scattered Linear Orderings

In a preceding paper (Bruyère and Carton, automata on linear orderings, MFCS'01), automata have been introduced for words indexed by linear orderings. These automata are a generalization of automata for finite, infinite, bi-infinite and even transfinite words studied by Büchi. Kleene's theorem has been generalized to these words. We prove that rational sets of words on countable scattered linear orderings are closed under complementation using an algebraic approach.

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Complementation of rational sets on scattered linear orderings of finite rank

Theoretical Computer Science ◽

10.1016/j.tcs.2007.03.008 ◽

2007 ◽

Vol 382 (2) ◽

pp. 109-119

Author(s):

Olivier Carton ◽

Chloé Rispal

Keyword(s):

Finite Rank ◽

Linear Orderings ◽

Rational Sets

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Complementation of Rational Sets on Scattered Linear Orderings of Finite Rank

LATIN 2004: Theoretical Informatics - Lecture Notes in Computer Science ◽

10.1007/978-3-540-24698-5_33 ◽

2004 ◽

pp. 292-301 ◽

Cited By ~ 2

Author(s):

Olivier Carton ◽

Chloé Rispal

Keyword(s):

Finite Rank ◽

Linear Orderings ◽

Rational Sets

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COMPLETELY REDUCIBLE SETS

International Journal of Algebra and Computation ◽

10.1142/s0218196713400158 ◽

2013 ◽

Vol 23 (04) ◽

pp. 915-941 ◽

Cited By ~ 4

Author(s):

DOMINIQUE PERRIN

Keyword(s):

Closure Properties ◽

Linear Representations ◽

Complete Reducibility ◽

The Family ◽

Syntactic Representation ◽

Completely Reducible ◽

Rational Sets ◽

Cyclic Sets

We study the family of rational sets of words, called completely reducible and which are such that the syntactic representation of their characteristic series is completely reducible. This family contains, by a result of Reutenauer, the submonoids generated by bifix codes and, by a result of Berstel and Reutenauer, the cyclic sets. We study the closure properties of this family. We prove a result on linear representations of monoids which gives a generalization of the result concerning the complete reducibility of the submonoid generated by a bifix code to sets called birecurrent. We also give a new proof of the result concerning cyclic sets.

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