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It is shown that the Hecke–Kiselman algebra associated to a finite directed graph is an automaton algebra in the sense of Ufnarovskii. Consequently, its Gelfand–Kirillov dimension is an integer if it is finite. Moreover, it is proved that the Hecke–Kiselman algebra associated to an oriented cycle admits a finite Gröbner basis.

Gröbner basis and the automaton property of Hecke–Kiselman algebras