The class of row monomial matrices (one unit and rest zeros in every row) with some non-standard operations of summation and usual multiplication is our main object. These matrices generate a space with respect to the mentioned operations. A word $w$ of letters on edges of underlying graph $\Gamma$ of deterministic finite automaton (DFA) is called synchronizing if $w$ sends all states of the automaton to a unique state ($|R(w)|=1$. J. \v{C}erny discovered in 1964 a sequence of $n$-state complete DFA possessing a minimal synchronizing word of length $(n-1)^2$. The hypothesis, well known today as the \v{C}erny conjecture, claims that $(n-1)^2$ is also precise upper bound on the length of such a word for a complete DFA. The hypothesis was formulated in 1966 by Starke. The problem has motivated great and constantly growing number of investigations and generalizations. The proof of the conjecture is based on connection between length of words $u$ and dimension of the space generated by row monomial matrices $M_u$, the set of synchronizing matrices placed some role.
翻译:以非标准操作方式进行总和和和通常倍增的一行单体矩阵(每行一个单位和零零休息)类别是我们的主要对象。 这些矩阵生成了一个与上述操作有关的空间。 在基本图形($\Gamma$)的边缘, 确定性非自闭式自动数学(DFA) 的字母值为1美元, 如果美元将自动磁盘的所有状态发送到一个独特的状态($R(w)+1美元) 。 J.\v{C}erny 于1964年发现一连串美元状态完整的 DFA, 含有一个最小同步的字数(n-1) 2美元。 假设, 现今称为 $\ {C} ERjecture, 声称$( n-1) 2美元也是该词长度的精确上限, 用于完整的 DFAFA。 假设于1966年由 Starke制定。 问题促使大量且不断增加的调查和概括性。 预测性的证据依据了美元字数的长度、 美元 和 美元 基质 所设定的空间同步的基质 之间的连接。