Geodesic
Algebraic structures on double and plane posets
Journal of Algebraic Combinatorics, Tome 37 (2013) no. 1, pp. 39-66.

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Summary: We study the Hopf algebra of double posets and two of its Hopf subalgebras, the Hopf algebras of plane posets and of posets "without N". We prove that they are free, cofree, self-dual, and we give an explicit Hopf pairing on these Hopf algebras. We also prove that they are free 2-As algebras; in particular, the Hopf algebra of posets "without N" is the free 2-As algebra on one generator. We deduce a description of the operads of 2-As algebras and of B $_{\infty }$ algebras in terms of plane posets.
Keywords: combinatorial Hopf algebras, 2-as algebras, double posets, plane posets 
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     title = {Algebraic structures on double and plane posets},
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Foissy, Loïc. Algebraic structures on double and plane posets. Journal of Algebraic Combinatorics, Tome 37 (2013) no. 1, pp. 39-66. http://geodesic.mathdoc.fr/item/JAC_2013__37_1_a5/