Geodesic
Normal form in Hecke-Kiselman monoids associated with simple oriented graphs
Algebra and discrete mathematics, Tome 30 (2020) no. 2, pp. 161-171

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We generalize Kudryavtseva and Mazorchuk's concept of a canonical form of elements [9] in Kiselman's semigroups to the setting of a Hecke-Kiselman monoid $\mathbf{HK}_\Gamma$ associated with a simple oriented graph $\Gamma$. We use confluence properties from [7] to associate with each element in $\mathbf{HK}_\Gamma$ a normal form; normal forms are not unique, and we show that they can be obtained from each other by a sequence of elementary commutations. We finally describe a general procedure to recover a (unique) lexicographically minimal normal form.
Keywords: simple oriented graph, Hecke-Kiselman monoid, normal form. 
@article{ADM_2020_30_2_a0,
     author = {R. Aragona and A. D'Andrea},
     title = {Normal form in {Hecke-Kiselman} monoids associated with simple oriented graphs},
     journal = {Algebra and discrete mathematics},
     pages = {161--171},
     publisher = {mathdoc},
     volume = {30},
     number = {2},
     year = {2020},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ADM_2020_30_2_a0/}
}
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R. Aragona; A. D'Andrea. Normal form in Hecke-Kiselman monoids associated with simple oriented graphs. Algebra and discrete mathematics, Tome 30 (2020) no. 2, pp. 161-171. http://geodesic.mathdoc.fr/item/ADM_2020_30_2_a0/