论文标题
循环基团的有限最大代码和因素化
Finite maximal codes and factorizations of cyclic groups
论文作者
论文摘要
可变长度代码是游离单体的自由下monomoid的基础。关于有限最大代码的结构，即分解猜想和三角形猜想，有一些重要的长期开放问题，这是Perrin和Schützemberger提出的。后者涉及有限代码$ y $，是$ a^* b a^* $的子集，其中$ a $是字母，$ b $是一个不包含$ a $的字母。 Zhang和Shum最近显示了有限最大代码的结构特性。它表现出有限的最大代码与循环基团的因素化之间的关系。为了突出显示此结果与其他较旧的链接在最大和分解代码上之间的链接，我们为此结果提供了更简单，更新的证明。结果，我们证明，对于任何有限的最大代码$ x \ subseteq（b \ cup \ {a \}）^*$包含单词$ a^{pq} $，其中$ p，q $是prime数字，$ x \ cap a^*b a^*b a^*$满足三角形的猜想。让$ n $成为一个正整数，最多是两个质数的产物。我们还证明，是否有限代码$ y \ cup a^{n} \ subseteq a^* b a^* \ cup a^* $包含在有限的最大代码中，并且如果保留，则包括$ y \ y \ y \ cup a^{n} $也包含在某个代码中，也可以满足因素化的建议。
Variable-length codes are the bases of the free submonoids of a free monoid. There are some important longstanding open questions about the structure of finite maximal codes, namely the factorization conjecture and the triangle conjecture, proposed by Perrin and Schützemberger. The latter concerns finite codes $Y$ which are subsets of $a^* B a^*$, where $a$ is a letter and $B$ is an alphabet not containing $a$. A structural property of finite maximal codes has recently been shown by Zhang and Shum. It exhibits a relationship between finite maximal codes and factorizations of cyclic groups. With the aim of highlighting the links between this result and other older ones on maximal and factorizing codes, we give a simpler and a new proof of this result. As a consequence, we prove that for any finite maximal code $X \subseteq (B \cup \{a \})^*$ containing the word $a^{pq}$, where $p,q$ are prime numbers, $X \cap a^* B a^*$ satisfies the triangle conjecture. Let $n$ be a positive integer that is a product of at most two prime numbers. We also prove that it is decidable whether a finite code $Y \cup a^{n} \subseteq a^* B a^* \cup a^*$ is included in a finite maximal code and that, if this holds, $Y \cup a^{n}$ is included in a code that also satisfies the factorization conjecture.
