A fruitful idea in combinatorics is to consider the totality of objects of a certain type, and encode natural operations on these objects in operations of some algebra with coefficients in a field of characteristic 0. We introduce a general class of combinatorial objects, which we call multi-complexes, which simultaneously generalizes graphs, multigraphs, hypergraphs and simplicial and delta complexes. A natural algebra of multi-complexes is defined as the vector space which has a formal basis C consisting of all isomorphism types of multi-complexes, and multiplication the disjoint union. This is a Hopf algebra with an operation encoding assembly and disassembly information for such objects, and extends and generalizes the Hopf algebra of graphs. Such Hopf algebras are connected, graded commutative and cocommutative, and by general results of Cartier-Kostant-Milnor-Moore, are just a polynomial algebra. However, it comes with additional structure which encodes combinatorial information, and the main goal is to describe its structure in a way relating to combinatorics. Such structures have been of high interest in literature, both intrinsically and due to applications to combinatorial problems.
We give a solution to the problem in this generality and find an explicit basis B of the space of primitives, and which is of combinatorial relevance: it is such that each multi-complex is a polynomial with non-negative integer coefficients of the elements of B, and each element of B is a polynomial with integer coefficients in C. The coefficients appearing in all these polynomials are, up to signs, numbers counting multiplicities of sub-multi-complexes in a multi-complex. Using this, we also solve the antipode formula problem, and find the cancellation and grouping free formula for the antipode. Such formulas are usually hard to obtain, and constitute a main question for such algebras
The main method is to use certain Mobius type formulas for posets of
sub-objects of multi-complexes. As examples, we will explicitly
illustrate how our results specialize to the formulas for graphs or
simplicial complexes, and relations of these to open questions such
as the graph reconstruction conjecture.