On incidence algebras description of cobweb posets

The explicite formulas for Mobius function and some other important elements of the incidence algebra of an arbitrary cobweb poset are delivered. For that to do one uses Kwasniewski’s construction of his cobweb posets . The digraph representation of these cobweb posets constitutes a newly discovered class of orderable DAG’s named here down KoDAGs with a kind of universality now being investigated. Namely cobweb posets’ and thus KoDAGs’s defining di-bicliques are links of any complete relations’ chains.

💡 Research Summary

The paper investigates cobweb posets—layered partially ordered sets defined by an arbitrary integer sequence {Fₙ}—through the lens of incidence algebras and directed acyclic graph (DAG) theory. Each level Lₙ contains Fₙ elements, and every element of Lₙ is comparable to every element of the next level Lₙ₊₁. This complete bipartite relationship is modeled as a di‑biclique K_{Fₙ,Fₙ₊₁}. By chaining these di‑bicliques, the authors construct a visual representation of the whole poset as a DAG whose edges are precisely the links between successive levels. They name this class of order‑able DAGs “KoDAGs”.

The incidence algebra I(Π) of a poset Π consists of functions defined on intervals