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- W4300960216 abstract "We initiate the study of a large class of species monoids and comonoids which come equipped with a poset structure that is compatible with the multiplication and comultiplication maps. We show that if a monoid and a comonoid are related through a Galois connection, then they are dual to each other. This duality is best understood by introducing a new basis constructed through Mobius inversion. We use this new basis to give uniform proofs for cofreeness and calculations of primitives for the Hopf monoids of set partitions, graphs, hypergraphs, and simplicial complexes. Further, we show that the monoid and comonoid of a Hopf monoid are related through a Galois connection if and only if the Hopf monoid is linearized, commutative, and cocommutative. In these cases, we give a grouping-free formula for the antipode in terms of an evaluation of the characteristic polynomial of a related poset. This gives new proofs for the antipodes of the Hopf monoids of graphs, hypergraphs, set partitions, and simplicial complexes." @default.
- W4300960216 created "2022-10-04" @default.
- W4300960216 creator A5033013913 @default.
- W4300960216 date "2020-05-27" @default.
- W4300960216 modified "2023-09-27" @default.
- W4300960216 title "Poset Hopf Monoids" @default.
- W4300960216 doi "https://doi.org/10.48550/arxiv.2005.13707" @default.
- W4300960216 hasPublicationYear "2020" @default.
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