اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدBirecurrent sets
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A set is called recurrent if its minimal automaton is strongly connected and birecurrent if it is recurrent as well as its reversal. We prove a series of results concerning birecurrent sets. It is already known that any birecurrent set is completely reducible (that is, such that the minimal representation of its characteristic series is completely reducible). The main result of this paper characterizes completely reducible sets as linear combinations of birecurrent sets
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In this note we study automata recognizing birecurrent sets. A set of words is birecurrent if the minimal partial DFA recognizing this set and the minimal partial DFA recognizing the reversal of this set are both strongly connected. This notion was i
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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.
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e does not exist another submonoid generated by at most $k$ words containing $M$. We call a set $X subseteq A^*$ primitive if it is the basis of a $|X|$-maximal submonoid. This extends the notion of primitive word: indeed, ${w}$ 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 $X subseteq Y^*$. The set $Y$ is therefore called 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. This result cannot be extended to sets of rank larger than 2. For a single word $w$, we say that the set ${x,y}$ is a {em binary 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 binary root ${x,y}$ such that $|x|+|y|<sqrt{|w|}$. That is, the binary root of a word is unique provided the length of the word is sufficiently large with respect to the size 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.
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