A graph-dynamical interpretation of Kiselman's semigroups

A Sequential Dynamical System (SDS) is a quadruple (\Gamma, S_i,f_i,w) consisting of a (directed) graph \Gamma=(V,E), each of whose vertices i\in V is endowed with a finite set state S_i and an update function f_i: \prod_{j, i \to j} S_j \to S_i --- we call this structure an {\em update system} --- and a word w in the free monoid over V, specifying the order in which update functions are to be performed. Each word induces an evolution of the system and in this paper we are interested in the dynamics monoid, whose elements are all possible evolutions. When \Gamma is a directed acyclic graph, the dynamics monoid of every update system supported on \Gamma naturally arises as a quotient of the Hecke-Kiselman monoid associated with \Gamma. In the special case where \Gamma = \Gamma_n is the complete oriented acyclic graph on n vertices, we exhibit an update system whose dynamics monoid coincides with Kiselman's semigroup K_n, thus showing that the defining Hecke-Kiselman relations are optimal in this situation. We then speculate on how these results may extend to the general acyclic case.