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Register a new userTransfinite Lyndon words
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Added by
Olivier Carton
Publication date
2018
fields
Informatics Engineering
and research's language is
English
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In this paper, we extend the notion of Lyndon word to transfinite words. We prove two main results. We first show that, given a transfinite word, there exists a unique factorization in Lyndon words that are densely non-increasing, a relaxation of the condition used in the case of finite words. In the annex, we prove that the factorization of a rational word has a special form and that it can be computed from a rational expression describing the word.
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A generalized lexicographical order on infinite words is defined by choosing for each position a total order on the alphabet. This allows to define generalized Lyndon words. Every word in the free monoid can be factorized in a unique way as a nonincreasing factorization of generalized Lyndon words. We give new characterizations of the first and the last factor in this factorization as well as new characterization of generalized Lyndon words. We also give more specific results on two special cases: the classical one and the one arising from the alternating lexicographical order.
In this paper we compare two finite words $u$ and $v$ by the lexicographical order of the infinite words $u^omega$ and $v^omega$. Informally, we say that we compare $u$ and $v$ by the infinite order. We show several properties of Lyndon words expressed using this infinite order. The innovative aspect of this approach is that it allows to take into account also non trivial conditions on the prefixes of a word, instead that only on the suffixes. In particular, we derive a result of Ufnarovskij [V. Ufnarovskij, Combinatorial and asymptotic methods in algebra, 1995] that characterizes a Lyndon word as a word which is greater, with respect to the infinite order, than all its prefixes. Motivated by this result, we introduce the prefix standard permutation of a Lyndon word and the corresponding (left) Cartesian tree. We prove that the left Cartesian tree is equal to the left Lyndon tree, defined by the left standard factorization of Viennot [G. Viennot, Alg`ebres de Lie libres et monoides libres, 1978]. This result is dual with respect to a theorem of Hohlweg and Reutenauer [C. Hohlweg and C. Reutenauer, Lyndon words, permutations and trees, 2003].
A Lyndon word is a non-empty word strictly smaller in the lexicographic order than any of its suffixes, except itself and the empty word. In this paper, we show how Lyndon words can be used in the distributed control of a set of n weak mobile robots. By weak, we mean that the robots are anonymous, memoryless, without any common sense of direction, and unable to communicate in an other way than observation. An efficient and simple deterministic protocol to form a regular n-gon is presented and proven for n prime.
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 $X subseteq F^*$. We say that a submonoid $M$ generated by $k$ elements of $A^*$ is {em $k$-maximal} if there does not exist another submonoid generated by at most $k$ words containing $M$. We call a set $X subseteq A^*$ {em 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 $X subseteq Y^*$. We therefore call $Y$ a {em 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 {em 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 $|x|+|y|<sqrt{|w|}$. 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 $theta$ is defined on $A^*$. In this setting, the notions of $theta$-power, $theta$-primitive and $theta$-root are defined, and it is shown that any word has a unique $theta$-primitive root. This result can be obtained with our approach by showing that a word $w$ is $theta$-primitive if and only if ${w, theta(w)}$ is a primitive set.
We have reviewed some results on quantized shuffling, and in particular, the grading and structure of this algebra. In parallel, we have summarized certain details about classical shuffle algebras, including Lyndon words (primes) and the construction of bases of classical shuffle algebras in terms of Lyndon words. We have explained how to adapt this theory to the construction of bases of quantum group algebras in terms of Lyndon words. This method has a limited application to the specific case of the quantum group parameter being a root of unity, with the requirement that specialization to the root of unity is non-restricted. As an additional, applied part of this work, we have implemented a Wolfram Mathematica package with functions for quantum shuffle multiplication and constructions of bases in terms of Lyndon words.
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