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arXiv · 1407.0476

The Hopf algebra of finite topologies and T-partitions

Loïc Foissy·Claudia Malvenuto
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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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Loïc Foissy, Claudia Malvenuto. 2014-10-04. The Hopf algebra of finite topologies and T-partitions. https://arxiv.org/abs/1407.0476

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