[Paper Review] Natural join construction of graded posets versus ordinal sum and discrete hyper boxes

A. K. Kwaśniewski|ArXiv.org|Jul 15, 2009

Advanced Combinatorial Mathematics37 references3 citations

TL;DR

This paper introduces the natural join operation $P \oplus\!\to Q$ as a novel construction for graded posets, providing a unified framework that generalizes the ordinal sum and enables a simple derivation of the Möbius function formula for cobweb posets. The method leverages the natural join of bi-adjacency matrices of graded posets, interpreted as KoDAGs (Ko-complete DAGs), and establishes explicit formulas for Whitney numbers and characteristic polynomials using $F$-graded structures and $F$-nomials.

ABSTRACT

One introduces here the natural join $P \os Q$ of graded posets $< P,\leq_P >$ and $< Q,\leq_Q >$ with correspondingly maximal and minimal sets being identical as expressed by ordinal sum $P\oplus Q$ apart from other definition and due to that one arrives at a simple proof of the $M{ö}bius $ function formula for cobweb posets. We also quote the other authors explicit formulas for the zeta matrix and its inverse for any graded posets with the finite set of minimal elements from earlier works of the author. These formulas are based on the formulas for cobweb posets and their $Hasse$ diagrams or graphs named $KoDAGs$ which are interpreted as chains of binary complete or universal relations joined by the natural join operation. Natural join of two independent sets is therefore the ordinal sum of this trivially ordered posets represented also by directed biclique named dibiclique and correspondingly by their $Hasse $ diagrams or graphs named $KoDAGs$. Such cobweb posets and equivalently their Hasse diagrams or graphs named $KoDAGs$ are also encoded by discrete hyper-boxes and the natural join operation of such discrete hyper boxes is just cartesian product of them accompanied with projection out of common faces. All graded posets with no mute vertices in their $Hasse$ diagrams which means that no vertex has indegree or outdegree equal zero are natural join of chain of relations and may be at the same time interpreted an $n-ary$ relation, $n \in N \cup \{\infty \}$.

Motivation & Objective

• To unify and generalize the ordinal sum construction for graded posets using the natural join operation.
• To provide a simple, systematic proof of the Möbius function formula for cobweb posets via natural join and KoDAG representations.
• To derive explicit formulas for Whitney numbers and characteristic polynomials of cobweb posets using $F$-graded structures.
• To clarify the conditions under which $F$-graded posets can be represented as $n$-ary relations, excluding mute vertices.
• To establish connections between discrete hyper-boxes, natural join, and Cartesian product with projection of common faces.

Proposed method

• Define the natural join $P \oplus\!\to Q$ of two graded posets $P$ and $Q$ as a construction that identifies the maximal set of $P$ with the minimal set of $Q$, generalizing the ordinal sum.
• Represent cobweb posets as KoDAGs (Ko-complete DAGs), which are directed acyclic graphs encoding chains of binary complete relations.
• Use bi-adjacency matrices of successive levels to model the natural join operation, ensuring no zero rows or columns (no mute vertices) for $n$-ary relation representation.
• Derive the zeta matrix $[Max]$ as a power series $ (I - \kappa)^{-1} = \sum_{k \geq 0} \kappa^k $, where $\kappa$ is the incidence matrix.
• Construct the inverse zeta matrix $[Max]^{-1} = \delta - \kappa$ as a lower-triangular matrix with identity blocks and negative bi-adjacency matrices.
• Express the number of maximal chains in intervals $[x_k, x_n]$ via $[Max]_{k,n}$, leading to explicit formulas involving $F$-nomials and factorials.

Experimental results

Research questions

• RQ1 How does the natural join operation generalize the ordinal sum in the context of graded posets with identical maximal/minimal sets?
• RQ2 What is the role of KoDAGs and bi-adjacency matrices in representing cobweb posets and enabling Möbius function derivation?
• RQ3 Under what conditions can an $F$-graded poset be represented as an $n$-ary relation, and how does the absence of mute vertices affect this?
• RQ4 What explicit formulas can be derived for Whitney numbers and characteristic polynomials of cobweb posets using the natural join construction?
• RQ5 How do discrete hyper-boxes relate to the natural join and Cartesian product of graded posets?

Key findings

• The natural join $P \oplus\!\to Q$ provides a unified construction that generalizes the ordinal sum and allows for a simple derivation of the Möbius function formula for cobweb posets.
• The number of maximal chains in the interval $[x_k, x_n]$ is given by $[Max]_{k,n} = \binom{n-1}{k-2}_F (n-k+1)_F!$, where $\binom{n-1}{k-2}_F$ is an $F$-nomial coefficient.
• The inverse zeta matrix $[Max]^{-1}$ is a lower-triangular matrix with identity blocks on the diagonal and negative bi-adjacency matrices below, enabling efficient Möbius function computation.
• Cobweb posets with no mute vertices (no zero rows/columns in bi-adjacency matrices) can be interpreted as $n$-ary relations via the natural join of binary complete relations.
• The characteristic polynomial and Whitney numbers of cobweb posets are explicitly expressed using $F$-nomials and $F$-factorials derived from the natural join structure.
• Discrete hyper-boxes correspond to the natural join of $F$-graded posets, realized as Cartesian products with projection of common faces, preserving the poset structure.