Algebraic structures on double and plane posets (Q1938791)

The Hopf algebra \(H_{\mathrm{DP}}\) of isomorphism classes of double posets was introduced by \textit{C. Malvenuto} and \textit{C. Reutenauer} [J. Comb. Theory, Ser. A 118, No. 4, 1322--1333 (2011; Zbl 1231.05292)]. A double poset is a finite set with two partial orders. Two associative products are defined on DP, the set of isoclasses of double posets. Let \(P\) and \(Q\) be in DP. Both products are the disjoint union of \(P\) and \(Q\). One product requires that if \(x\) is in \(P\) and \(y\) in \(Q\), then \(x\) is less than or equal to \(y\) in the first product of \(P\) and \(Q\) and not comparable in the second. The other product is symmetrically defined. The author defines indecomposability for each of the two products of double posets and proves that as a 2-As algebra (i.e., as an algebra with two associative products sharing the same unit), \(H_{\mathrm{DP}}\) is free on the set of elements which are indecomposable with respect to one of the partial orders on it. Then the author studies plane posets, i.e., double posets in which two distinct elements are comparable in one partial order if and only if they are not comparable in the other partial order. The plane posets PP are those which do not contain three specific double posets. \(H_{\mathrm{PP}}\) is a sub-2-AS algebra of \(H_{\mathrm{DP}}\), and is also free on the plane posets indecomposable with respect to one of the two partial orders. Then the author studies WN-double posets. These are the plane posets which do not contain two specific subposets of four elements. The author shows that as a 2-AS algebra, \(H_{\mathrm{WNP}}\) is freely generated by the one-element double poset. A Hopf algebra structure on \(H_{\mathrm{DP}}\) is given using ideals \(I\) of a double poset \(P\), i.e., if \(x\) in \(I\) is less than \(y\) in \(P\), then \(y\) is in \(I\). Then the coproduct of \(P\) is the sum of the tensor products of \(P\backslash I\) and \(I\) over all ideals \(I\) of \(P\). \(H_{\mathrm{PP}}\) and \(H_{\mathrm{WNP}}\) are Hopf subalgebras of \(H_{\mathrm{DP}}\), and all three are free and cofree. A plane forest is a plane poset whose Hasse graph is a rooted forest. \(H_{\mathrm{PF}}\) is the co-opposite of the well-known Connes-Kreimer Hopf algebra of plane trees.