TITLE:
Algebra and Coalgebra on Posets
AUTHORS:
Rongrong Yuan, Huilan Li
KEYWORDS:
Conjunction Product, Unshuffle Coproduct, Poset, Graded, Free
JOURNAL NAME:
Open Journal of Applied Sciences,
Vol.12 No.7,
July
19,
2022
ABSTRACT: A lot of combinatorial objects have algebra and coalgebra structures and
posets are important combinatorial objects. In this paper, we construct algebra
and coalgebra structures on the vector space spanned by posets. Firstly, by
associativity and the unitary property, we prove that the vector space with the
conjunction product is a graded algebra. Then by the definition of free
algebra, we prove that the algebra is free. Finally, by the coassociativity and the counitary
property, we prove that the vector space with the unshuffle coproduct is a
graded coalgebra.