The Hopf algebra of finite topologies and T-partitions

2014/07/02 by Loïc Foissy, Foissy, Loïc, Claudia Malvenuto +1 · 1 citation
Mathematics · #FOS: Mathematics #Rings and Algebras (math.RA) #math.RA

paper · pdf · doi:10.48550/arxiv.1407.0476

33 pages. Second version, a few typos corrected

arxiv created 2014/10/04 · arxiv updated 2014/10/07

Abstract

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.

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• Operads and bialgebras of multi-indices, and Novikov algebras

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