On the conjugation of Standard morphisms

Séébold, Patrice

doi:10.1007/3-540-61550-4_174

Cited by 3 publications

(3 citation statements)

References 27 publications

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“…We apply Séébold's theory of conjugate morphisms, see [22], Sect. 2.3.4 or [29]: given two (endo)morphisms f , g of the free monoid {a, b} * , we say that g is a right conjugate of f if there is w ∈ {a, b} * such that f (x)w = wg(x) for all x ∈ {a, b}. This is clearly equivalent to the following property: there is a chain for the above-defined binary relation → from the couple ( f (a), f (b)) to the couple (g(a), g(b)).…”

Section: Chains Of Conjugate Bases Application To Palindromicitymentioning

confidence: 98%

Sturmian morphisms, the braid group B 4, Christoffel words and bases of F 2

Kassel

Reutenauer

2006

Annali di Matematica

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We give a presentation by generators and relations of a certain monoid generating a subgroup of index two in the group Aut(F 2 ) of automorphisms of the rank two free group F 2 and show that it can be realized as a monoid in the group B 4 of braids on four strings. In the second part we use Christoffel words to construct an explicit basis of F 2 lifting any given basis of the free abelian group Z 2 . We further give an algorithm allowing to decide whether two elements of F 2 form a basis or not. We also show that, under suitable conditions, a basis has a unique conjugate consisting of two palindromes.

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Section: Chains Of Conjugate Bases Application To Palindromicitymentioning

confidence: 98%

Sturmian morphisms, the braid group B 4, Christoffel words and bases of F 2

Kassel

Reutenauer

2006

Annali di Matematica

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“…The authors wish to express their gratitude to the anonymous referees for their valuable comments and suggestions and for bringing references [16,18,17] to their attention. This research was supported by Grant NSC 96-2115-M-033-005-MY3, the National Science Council, Republic of China.…”

Section: Acknowledgementsmentioning

confidence: 99%

“…In [18], Séébold introduced the notion of conjugate morphism, and proved that, if x is a standard word and H 0 is a standard morphism such that x = H 0 (a), then the set {H(a) : H is a conjugate of H 0 } equals the set of conjugates of x. In [17], Richomme developed some algorithms computing the conjugates of a standard morphism.…”

Section: Introductionmentioning

confidence: 99%

α-words and the radix order

Chuan

Chen

et al. 2011

Theoretical Computer Science

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a b s t r a c tLet α = (a 1 , a 2 , . . .) be a sequence (finite or infinite) of integers with a 1 ≥ 0 and a n ≥ 1, for all n ≥ 2. Let {a, b} be an alphabet. For n ≥ 1, and r = r 1 r 2 · · · r n ∈ N n , with 0 ≤ r i ≤ a i for 1 ≤ i ≤ n, there corresponds an nth-order α-word u n [r] with label r derived from the pair (a, b). These α-words are defined recursively as follows:Many interesting combinatorial properties of α-words have been studied by Chuan. In this paper, we obtain some new methods of generating the distinct α-words of the same order in lexicographic order. Among other results, we consider another function r  → w[r] from the set of labels of α-words to the set of α-words. The string r is called a new label of the α-word w[r].Using any new label of an nth-order α-word w, we can compute the number of the nthorder α-words that are less than w in the lexicographic order. With the radix orders < r on N n (regarding N as an alphabet) and {a, b} + with a < r b, we prove that there exists a subset D of the set of all labels such that w[r] < r w[s] whenever r, s ∈ D and r < r s.

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Tilings and quasiperiodicity

Durand

1997

Automata, Languages and Programming

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On the conjugation of Standard morphisms

Cited by 3 publications

References 27 publications

Sturmian morphisms, the braid group B 4, Christoffel words and bases of F 2

Sturmian morphisms, the braid group B 4, Christoffel words and bases of F 2

α-words and the radix order

Tilings and quasiperiodicity

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