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

Voir la notice de l'article provenant de la source Math-Net.Ru

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/}
}
TY  - JOUR
AU  - R. Aragona
AU  - A. D'Andrea
TI  - Normal form in Hecke-Kiselman monoids associated with simple oriented graphs
JO  - Algebra and discrete mathematics
PY  - 2020
SP  - 161
EP  - 171
VL  - 30
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/ADM_2020_30_2_a0/
LA  - en
ID  - ADM_2020_30_2_a0
ER  - 
%0 Journal Article
%A R. Aragona
%A A. D'Andrea
%T Normal form in Hecke-Kiselman monoids associated with simple oriented graphs
%J Algebra and discrete mathematics
%D 2020
%P 161-171
%V 30
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/ADM_2020_30_2_a0/
%G en
%F ADM_2020_30_2_a0
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/

[1] R. Aragona and A. D'Andrea, “Hecke–Kiselman monoids of small cardinality”, Semigroup Forum, 86 (2013), 32–40 | DOI | MR | Zbl

[2] C. L. Barrett, H. S. Mortveit, and C. M. Reidys, “Elements of a theory of simulation II: Sequential dynamical systems”, Appl. Math. Comput., 107 (2000), 121–136 | MR | Zbl

[3] E. Collina and A. D'Andrea, “A graph-dynamical interpretation of Kiselman's semigroups”, J. Alg. Comb., 41 (2015), 1115–1132 | DOI | MR | Zbl

[4] L. Forsberg, Effective representations of Hecke-Kiselman monoids of type A, preprint, arXiv: 1205.0676v4

[5] O. Ganyushkin and V. Mazorchuk, “On Kiselman quotients of 0-Hecke monoids”, Int. Electron. J. Algebra, 10 (2011), 174–191 | MR | Zbl

[6] F. Hivert, A. Schilling, N. M. Thiéry, “The biHecke monoid of a finite Coxeter group”, 22nd International Conference on Formal Power Series and Algebraic Combinatorics, FPSAC 2010, Discrete Math. Theor. Comput. Sci. Proc., 2010, 307–318 | MR | Zbl

[7] G. P. Huet, “Confluent Reductions: Abstract Properties and Applications to Term Rewriting Systems”, Journal of the ACM, 27 (1980), 797–821 | DOI | MR | Zbl

[8] C. O. Kiselman, “A semigroup of operators in convexity theory”, Trans. AMS, 354 (2002), 2035–2053 | DOI | MR | Zbl

[9] G. Kudryavtseva and V. Mazorchuk, “On Kiselman's semigroup”, Yokohama Math. J., 55 (2009), 21–46 | MR | Zbl

[10] A. Mȩcel and J. Okniński, “Growth alternative for Hecke–Kiselman monoids”, Publ. Mat., 63 (2019), 219–240 | DOI | MR

[11] A. Mȩcel and J. Okniński, “Gröbner basis and the automaton property of Hecke–Kiselman algebras”, Semigroup Forum, 99 (2019), 447–464 | DOI | MR

[12] M. H. A. Newman, “On theories with a combinatorial definition of “equivalence””, Ann. of Math., 43 (1942), 223–243 | DOI | MR | Zbl

[13] P. N. Norton, “$0$-Hecke algebras”, J. Austral. Math. Soc. (Series A), 27 (1979), 337–357 | DOI | MR

[14] J. Okniński, M. Wiertel, Combinatorics and structure of Hecke–Kiselman algebras, preprint, arXiv: 1904.12202 | MR

[15] R. Richardson and T. Springer, “The Bruhat order on symmetric varieties”, Geom. Dedic., 35 (1990), 389–436 | DOI | MR | Zbl