Hecke–Kiselman monoids [Formula: see text] and their algebras [Formula: see text], over a field [Formula: see text], associated to finite oriented graphs [Formula: see text] are studied. In the case [Formula: see text] is a cycle of length [Formula: see text], a hierarchy of certain unexpected structures of matrix type is discovered within the monoid [Formula: see text] and this hierarchy is used to describe the structure and the properties of the algebra [Formula: see text]. In particular, it is shown that [Formula: see text] is a right and left Noetherian algebra, while it has been known that it is a PI-algebra of Gelfand–Kirillov dimension one. This is used to characterize all Noetherian algebras [Formula: see text] in terms of the graphs [Formula: see text]. The strategy of our approach is based on the crucial role played by submonoids of the form [Formula: see text] in combinatorics and structure of arbitrary Hecke–Kiselman monoids [Formula: see text].

中文翻译：

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Hecke-Kiselman 代数的组合与结构

Hecke–Kiselman monoids [公式：参见文本] 及其代数 [公式：参见文本]，在域 [公式：参见文本] 上，与有限定向图 [公式：参见文本] 相关联进行了研究。在[公式：见文本]是一个长度循环[公式：见文本]的情况下，在幺半群[公式：见文本]中发现了某些意想不到的矩阵类型结构的层次结构，该层次结构用于描述结构以及代数的性质[公式：见正文]。特别是，[公式：见正文] 是一个左右诺特代数，而已知它是 Gelfand-Kirillov 维一维的 PI 代数。这用于根据图形 [公式：参见文本] 来表征所有诺特代数 [公式：参见文本]。