The Hopf algebra of finite topologies and T-partitions

Loïc Foissy (1) , Claudia Malvenuto (2)

1 LMPA - Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville
2 Sapienza University of Rome - Dipartimento di Matematica "Guido Castelnuovo" [Roma I]

A noncommutative and noncocommutative Hopf algebra on finite topologies H_T is introduced and studied (freeness, cofreeness, self-duality...). Generalizing Stanley's definition of P-partitions associated to a special poset, we define the notion of T-partitions associated to a finite topology, and deduce a Hopf algebra morphism from H_T to the Hopf algebra of packed words WQSym. Generalizing Stanley's decomposition by linear extensions, we deduce a factorization of this morphism, which induces a combinatorial isomorphism from the shuffle product to the quasi-shuffle product of WQSym. It is strongly related to a partial order on packed words, here described and studied.

Mots clés: Finite topologies; combinatorial Hopf algebras; packed words; partitions

Domaines: Anneaux et algèbres [math.RA]

Loïc Foissy, Claudia Malvenuto. The Hopf algebra of finite topologies and T-partitions. Journal of Algebra, 2015. ⟨hal-01017229v2⟩