The Hopf algebra of finite topologies and mould composition
Annales de l'Institut Fourier, Volume 67 (2017) no. 3, p. 911-945

We exhibit an internal coproduct on the Hopf algebra of finite topologies recently defined by the second author, C. Malvenuto and F. Patras, dual to the composition of “quasi-ormoulds”, which are the natural version of J. Ecalle’s moulds in this setting. All these results are displayed in the linear species formalism.

Nous mettons en évidence un coproduit interne sur l’algèbre de Hopf des topologies finies introduite récemment par C. Malvenuto, F. Patras et le second auteur. Ce coproduit est dual de la composition des “quasi-ormoules”, version naturelle des moules, selon la terminologie de J. Ecalle, dans ce contexte.

Received : 2016-03-19
Revised : 2016-09-16
Accepted : 2016-09-16
Published online : 2017-05-31
DOI : https://doi.org/10.5802/aif.3100
Classification:  05E05,  06A11,  16T30
Keywords: finite topological spaces, Hopf algebras, mould calculus, posets, quasi-orders
@article{AIF_2017__67_3_911_0,
     author = {Fauvet, Fr\'ed\'eric and Foissy, Lo\"\i c and Manchon, Dominique},
     title = {The Hopf algebra of finite topologies and mould composition},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {67},
     number = {3},
     year = {2017},
     pages = {911-945},
     doi = {10.5802/aif.3100},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2017__67_3_911_0}
}
Fauvet, Frédéric; Foissy, Loïc; Manchon, Dominique. The Hopf algebra of finite topologies and mould composition. Annales de l'Institut Fourier, Volume 67 (2017) no. 3, pp. 911-945. doi : 10.5802/aif.3100. http://www.numdam.org/item/AIF_2017__67_3_911_0/

[1] Aguiar, Marcelo; Mahajan, Swapneel Monoidal functors, species and Hopf algebras, Amer. Math. Soc., Providence, R.I., CRM Monographs Series, Tome 29 (2010), li+784 pages

[2] Aguiar, Marcelo; Mahajan, Swapneel Hopf monoids in the category of species, Proceedings of the international conference, University of Almeréa, Almeréa, Spain, July 4?8, 2011. (Contemporary Mathematics) Tome 585 (2013), pp. 17-124

[3] Aguiar, Marcelo; Santos, Walter Ferrer; Moreira, Walter The Heisenberg product: from Hopf algebras and species to symmetric functions (2015) (https://arxiv.org/abs/1504.06315 )

[4] Alexandroff, Pavel Diskrete Räume, Rec. Math. Moscou, n. Ser., Tome 2 (1937) no. 3, pp. 501-519

[5] Bergeron, Nantel; Zabrocki, Mike The Hopf algebras of symmetric functions and quasi-symmetric functions in non-commutative variables are free and co-free, J. Algebra Appl., Tome 8 (2009) no. 4, pp. 581-600 | Article

[6] Brown, Kenneth S. Semigroups, rings, and Markov chains, J. Theor. Probab., Tome 13 (2000) no. 3, pp. 871-938 | Article

[7] Calaque, Damien; Ebrahimi-Fard, Kurusch; Manchon, Dominique Two interacting Hopf algebras of trees, Adv. Appl. Math., Tome 47 (2011) no. 2, pp. 282-308 | Article

[8] Duchamp, Gérard; Hivert, Florent; Thibon, Jean-Yves Noncommutative symmetric functions VI: free quasi-symmetric functions and related algebras, Int. J. Algebra Comput., Tome 12 (2002) no. 5, pp. 671-717 | Article

[9] Ecalle, Jean Singularités non abordables par la géométrie, Ann. Inst. Fourier, Tome 42 (1992) no. 1-2, pp. 73-164 | Article

[10] Ecalle, Jean La trigèbre des ormoules (2010) (Private communication)

[11] Ecalle, Jean; Vallet, Bruno The arborification–coarborification transform: analytic, combinatorial, and algebraic aspects, Ann. Fac. Sci. Toulouse, Tome 13 (2004) no. 4, pp. 575-657 | Article

[12] Fauvet, Frédéric; Menous, Frédéric Ecalle’s arborification-coarborification transforms and Connes-Kreimer Hopf algebra (https://arxiv.org/abs/1212.4740, to appear in Ann. Sci. Éc. Norm. Supér. (4))

[13] Foissy, Loïc Algebraic structures on double and plane posets, J. Algebr. Comb., Tome 37 (2013) no. 1, pp. 39-66 | Article

[14] Foissy, Loïc Plane posets, special posets and permutations, Adv. Math., Tome 240 (2013), pp. 24-60 | Article

[15] Foissy, Loïc; Malvenuto, Claudia The Hopf algebra of finite topologies and 𝒯-partitions, J. Algebra, Tome 438 (2015), pp. 130-169 | Article

[16] Foissy, Loïc; Malvenuto, Claudia; Patras, Frédéric B -algebras, their enveloping algebras and finite spaces, J. Pure Appl. Algebra, Tome 220 (2016) no. 6, pp. 2434-2458 | Article

[17] Foissy, Loïc; Novelli, Jean-Christophe; Thibon, Jean-Yves Deformations of shuffles and quasi-shuffles, Ann. Inst. Fourier, Tome 66 (2016) no. 1, pp. 209-237 | Article

[18] Foissy, Loïc; Novelli, Jean-Christophe; Thibon, Jean-Yves Polynomial realizations of some combinatorial Hopf algebras, J. Noncommut. Geom., Tome 8 (2104) no. 1, pp. 141-162 | Article

[19] Gessel, Ira M. Multipartite P-partitions and inner products of skew Schur functions, Combinatorics and algebra, Boulder, Colorado, 1983, Amer. Math. Soc., Providence, R.I. (Contemp. Math.) Tome 34 (1984), pp. 289-317

[20] Hoffman, Michael E. Quasi-shuffle products, J. Algebr. Comb., Tome 11 (2000) no. 1, pp. 49-68 | Article

[21] Malvenuto, Claudia; Reutenauer, Christophe Duality between quasi-symmetric functions and the Solomon descent algebra, J. Algebra, Tome 177 (1995) no. 3, pp. 967-982 | Article

[22] Manchon, Dominique On bialgebras and Hopf algebras of oriented graphs, Confluentes Math., Tome 4 (2012) no. 1 (10 pp. (electronic)) | Article

[23] Menous, Frédéric An example of local analytic q-difference equation: analytic classification, Ann. Fac. Sci. Toulouse, Tome 15 (2006) no. 4, pp. 773-814 | Article

[24] Menous, Frédéric On the stability of some groups of formal diffeomorphisms by the Birkhoff decomposition, Adv. Math., Tome 216 (2007) no. 1, pp. 1-28 | Article

[25] Novelli, Jean-Christophe; Thibon, Jean-Yves Parking functions and descent algebras, Ann. Comb., Tome 11 (2007) no. 1, pp. 59-68 | Article

[26] Stanley, Richard P. Ordered structures and partitions, Mem. Am. Math. Soc., Tome 119 (1972) (104 pp.)

[27] Stanley, Richard P. Enumerative Combinatorics Vol. 2, Cambridge University Press, Cambridge Studies in Advanced Mathematics, Tome 62 (2001), xii+585 pages

[28] Stanley, Richard P. Enumerative Combinatorics Vol. 1, Cambridge University Press, Cambridge Studies in Advanced Mathematics, Tome 49 (2011), iv+642 pages

[29] Steiner, Anne K. The lattice of topologies: structure and complementation, Trans. Am. Math. Soc., Tome 122 (1966), pp. 379-398 | Article

[30] Vaidyanathaswamy, Ramaswamy S. Set topology, Chelsea, New-York (1960)