Antipodes and incidence coalgebras
Let \({\mathcal P}\) be a collection of locally finite partially ordered sets. An equivalence relation \(\sim\) on the collection of all segments of intervals in \({\mathcal P}\) is said to be order-compatible if [x,y]\(\sim [u,v]\) implies that there exists a bijection \(\phi: [x,y]\to [u,v]\) such that for all z in \([x,y],[x,z]\sim [\phi(x),\phi(z)]\) and \([z,y]\sim [\phi(z),\phi(y)].\) The (reduced) incidence coalgebra \(C({\mathcal P}_{\sim})\) modulo \(\sim\) is the free vector space spanned by the variables \(X^{\alpha}\), where \(\alpha\) is a type or equivalence class with comultiplication given by \(\Delta X^{\alpha}= \sum_{\sigma} X^{\sigma^-}\otimes X^{\sigma^+}\), where the sum on the right is computed by taking a representative [x,y] in \(\alpha\) and summing over all z, \(x\leq z\leq y\), with \(\sigma^-\) the type of [x,z] and \(\sigma^+\) the type of [z,y] [cf. \textit{J. R. Goldman} and \textit{G.-C. Rota}, Stud. Appl. Math. 49, 239-258 (1970; Zbl 0212.033)]. Among the results are: (a) An equivalence relation (containing all isomorphisms) is order-compatible if and only if, in the incidence algebra modulo isomorphism, the subspace spanned by \(X^ P-X^ Q\), where \(P\sim Q\), is a coideal; (b) construction of the free incidence bialgebra; (c) conditions for an incidence coalgebra to be embeddable combinatorially into an incidence Hopf algebra; (d) a recursion and a closed formula for the antipode of an incidence Hopf algebra. For enumeration over posets, the most important result is probably the expression for the antipode as a sum over alternating chains. As the author observes, since the antipode of H is the two-sided inverse of the identity map \(H\to H\) under convolution, the antipode is an analogue of the Möbius function [see \textit{G.-C. Rota}, Z. Wahrsch. Verw. Gebiete 2, 340-368 (1964)] and the closed formula is reminiscent of \textit{P. Hall}'s formula for the Möbius function [Q. J. Math., Oxf. II. Ser. 7, 134-151 (1936; Zbl 0014.10402)]. The results are illustrated by the binomial bialgebra constructed from Boolean algebras, the Faà di Bruno bialgebra constructed from partition lattices, and bialgebras constructed from hereditary classes of posets. In particular, the inversion formula of \textit{J. L. Lagrange} [Nouvelles Mém. Acad. Sci. Berlin 24 (1770); per revr.] is the antipode formula in the Faà di Bruno bialgebra.
scientific article; zbMATH DE number 4145902