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].
The Hecke-Kiselman algebra of a finite oriented graph Θ over a field K is studied. If Θ is an oriented cycle, it is shown that the algebra is semiprime and its central localization is a finite direct product of matrix algebras over the field of rational functions K(x). More generally, the radical is described in the case of PI-algebras, and it is shown that it comes from an explicitly described congruence on the underlying Hecke-Kiselman monoid. Moreover, the algebra modulo the radical is again a Hecke-Kiselman algebra and it is a finite module over its center.
Hecke-Kiselman algebras AΘ, over a field k, associated to finite oriented graphs Θ are considered. It has been known that every such algebra is an automaton algebra in the sense of Ufranovskii. In particular, its Gelfand-Kirillov dimension is an integer if it is finite. In this paper, a numerical invariant of the graph Θ that determines the dimension of AΘ is found. Namely, we prove that the Gelfand-Kirillov dimension of AΘ is the sum of the number of cyclic subgraphs of Θ and the number of oriented paths of a special type in the graph, each counted certain specific number of times.
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