The Hopf algebra of finite topologies and T-partitions

TL;DR

This paper introduces a noncommutative Hopf algebra on finite topologies, generalizes Stanley's P-partitions to T-partitions, and explores their algebraic and combinatorial properties, including a morphism to WQSym and a related partial order.

Contribution

It develops a new Hopf algebra structure on finite topologies and extends the concept of P-partitions to T-partitions, establishing connections with packed words and quasi-shuffle products.

Findings

Defined a noncommutative Hopf algebra on finite topologies.

Established a Hopf algebra morphism from H_T to WQSym.

Identified a partial order on packed words related to the algebraic structures.

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.