On -episturmian words (original) (raw)
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On different generalizations of episturmian words
Theoretical Computer Science, 2008
In this paper we study some classes of infinite words generalizing episturmian words, and analyse the relations occurring among such classes. In each case, the reversal operator R is replaced by an arbitrary involutory antimorphism ϑ of the free monoid A * . In particular, we define the class of ϑ-words with seed, whose "standard" elements (ϑ-standard words with seed) are constructed by an iterative ϑ-palindrome closure process, starting from a finite word u 0 called the seed. When the seed is empty, one obtains ϑ-words; episturmian words are exactly the R-words. One of the main theorems of the paper characterizes ϑ-words with seed as infinite words closed under ϑ and having at most one left special factor of each length n ≥ N (where N is some nonnegative integer depending on the word). When N = 0 we call such words ϑ-episturmian. Further results on the structure of ϑ-episturmian words are proved. In particular, some relationships between ϑ-words (with or without seed) and ϑ-episturmian words are shown.
A generalized palindromization map in free monoids
Theoretical Computer Science, 2012
The palindromization map ψ in a free monoid A * was introduced in 1997 by the first author in the case of a binary alphabet A, and later extended by other authors to arbitrary alphabets. Acting on infinite words, ψ generates the class of standard episturmian words, including standard Arnoux-Rauzy words. In this paper we generalize the palindromization map, starting with a given code X over A. The new map ψ X maps X * to the set PAL of palindromes of A * . In this way some properties of ψ are lost and some are saved in a weak form. When X has a finite deciphering delay one can extend ψ X to X ω , generating a class of infinite words much wider than standard episturmian words. For a finite and maximal code X over A, we give a suitable generalization of standard Arnoux-Rauzy words, called X-AR words. We prove that any X-AR word is a morphic image of a standard Arnoux-Rauzy word and we determine some suitable linear lower and upper bounds to its factor complexity.
On prefixal factorizations of words
European Journal of Combinatorics, 2016
We consider the class P 1 of all infinite words x ∈ A ω over a finite alphabet A admitting a prefixal factorization, i.e., a factorization x = U 0 U 1 U 2 · · · where each U i is a non-empty prefix of x. With each x ∈ P 1 one naturally associates a "derived" infinite word δ(x) which may or may not admit a prefixal factorization. We are interested in the class P ∞ of all words x of P 1 such that δ n (x) ∈ P 1 for all n ≥ 1. Our primary motivation for studying the class P ∞ stems from its connection to a coloring problem on infinite words independently posed by T. Brown in [3] and by the second author in . More precisely, let P be the class of all words x ∈ A ω such that for every finite coloring ϕ : A + → C there exist c ∈ C and a factorization x = V 0 V 1 V 2 · · · with ϕ(V i ) = c for each i ≥ 0. In [5] we conjectured that a word x ∈ P if and only if x is purely periodic. In this paper we show that P ⊆ P ∞ , so in other words, potential candidates to a counter-example to our conjecture are amongst the non-periodic elements of P ∞ . We establish several results on the class P ∞ . In particular, we show that a Sturmian word x belongs to P ∞ if and only if x is nonsingular, i.e., no proper suffix of x is a standard Sturmian word.
Square-free words on partially commutative free monoids
Information Processing Letters, 1986
We give a characterization of the partially commutative free monoids having an infinite number of square-free elements. We prove that it is decidable whether a given partially commutative free monoid contains infinitely many square-free words.
Quasiperiodic and Lyndon episturmian words
Theoretical Computer Science, 2008
Recently the second two authors characterized quasiperiodic Sturmian words, proving that a Sturmian word is non-quasiperiodic if and only if it is an infinite Lyndon word. Here we extend this study to episturmian words (a natural generalization of Sturmian words) by describing all the quasiperiods of an episturmian word, which yields a characterization of quasiperiodic episturmian words in terms of their directive words. Even further, we establish a complete characterization of all episturmian words that are Lyndon words. Our main results show that, unlike the Sturmian case, there is a much wider class of episturmian words that are non-quasiperiodic, besides those that are infinite Lyndon words. Our key tools are morphisms and directive words, in particular normalized directive words, which we introduced in an earlier paper. Also of importance is the use of return words to characterize quasiperiodic episturmian words, since such a method could be useful in other contexts.
Some generalizations of episturmian words and morphisms
We study some classes of infinite words that generalize standard episturmian words, defined by replacing the reversal operator with an arbitrary involutory antimorphism ϑ of A * . An analysis of the relations occurring among such classes of words, and of the morphisms connecting them to standard episturmian words, is given. In particular, we analyse some structural properties of standard ϑ-episturmian words and their characteristic morphisms.
arXiv (Cornell University), 2020
Given a (finite or infinite) subset X of the free monoid A * over a finite alphabet A, the rank of X is the minimal cardinality of a set F such that X ⊆ F *. We say that a submonoid M generated by k elements of A * is k-maximal if there does not exist another submonoid generated by at most k words containing M. We call a set X ⊆ A * primitive if it is the basis of a |X|-maximal submonoid. This definition encompasses the notion of primitive word-in fact, {w} is a primitive set if and only if w is a primitive word. By definition, for any set X, there exists a primitive set Y such that X ⊆ Y *. We therefore call Y a primitive root of X. As a main result, we prove that if a set has rank 2, then it has a unique primitive root. To obtain this result, we prove that the intersection of two 2-maximal submonoids is either the empty word or a submonoid generated by one single primitive word. For a single word w, we say that the set {x, y} is a bi-root of w if w can be written as a concatenation of copies of x and y and {x, y} is a primitive set. We prove that every primitive word w has at most one bi-root {x, y} such that |x| + |y| < |w|. That is, the bi-root of a word is unique provided the word is sufficiently long with respect to the size (sum of lengths) of the root. Our results are also compared to previous approaches that investigate pseudo-repetitions, where a morphic involutive function θ is defined on A *. In this setting, the notions of θ-power, θ-primitive and θ-root are defined, and it is shown that any word has a unique θ-primitive root. This result can be obtained with our approach by showing that a word w is θ-primitive if and only if {w, θ(w)} is a primitive set.
On the superimposition of Christoffel words
Theoretical Computer Science, 2011
Initially stated in terms of Beatty sequences, the Fraenkel conjecture can be reformulated as follows: for a k-letter alphabet A, with a fixed k ≥ 3, there exists a unique balanced infinite word, up to letter permutations and shifts, that has mutually distinct letter frequencies. Motivated by the Fraenkel conjecture, we study in this paper whether two Christoffel words can be superimposed. Following from previous works on this conjecture using Beatty sequences, we give a necessary and sufficient condition for the superimposition of two Christoffel words having same length, and more generally, of two arbitrary Christoffel words. Moreover, for any two superimposable Christoffel words, we give the number of different possible superimpositions and we prove that there exists a superimposition that works for any two superimposable Christoffel words. Finally, some new properties of Christoffel words are obtained as well as a geometric proof of a classic result concerning the money problem, using Christoffel words.
Infinite words without palindrome
2009
We show that there exists an uniformly recurrent infinite word whose set of factors is closed under reversal and which has only finitely many palindromic factors.
On a Product of Finite Monoids
Semigroup Forum, 1998
In this paper, for each positive integer m, we associate with a finite monoid S0 and m finite commutative monoids S 1 ,…, S m , a product ◊ m (S m ,…, S 1 , S 0). We give a representation of the free objects in the pseudovariety ◊ m (W m ,…, W 1 , W 0) generated by these (m + 1)-ary products where S i ∈ W i for all 0 ≤ i ≤ m. We then give, in particular, a criterion to determine when an identity holds in ◊ m (J 1 ,…,J 1 ,J 1) with the help of a version of the Ehrenfeucht-Fraïssé game (J 1 denotes the pseudovariety of all semilattice monoids). The union (J 1 ,…, J 1 , J 1) turns out to be the second level of the Straubing's dot-depth hierarchy of aperiodic monoids. Article: corresponding to the pseudovariety A. For a finite alphabet A, the class A * A is the least class of languages of A * (the free monoid generated by A) satisfying the following three conditions: A* A is closed under finite boolean operations, If L, L' ∈ A* A, then the concatenation LL' ∈ A* A, {u} ∈ A * A for all u ∈ A *. Cohen and Brzozowski [15] introduced the dot-depth hierarchy for the aperiodic languages of A + = A * \ {1} (1 denotes the empty word), and Straubing [31] defined another hierarchy for the aperiodic languages of A *. The Straubing hierarchy is a doubly indexed hierarchy for the class A * A. It grows out in a natural manner from the applications of concatenation and boolean operations. We proceed inductively starting with A * V 0 = {∅, A * }. Assuming that A*V k-1 is already defined for some k > 0, the class A * V k is defined as the boolean closure of all languages of the form L 0 a