[Paper Review] On natural join of posets properties and first applications

A. K. Kwaśniewski|ArXiv.org|Aug 10, 2009

Advanced Combinatorial Mathematics43 references3 citations

TL;DR

This paper introduces a new natural join operation $P \oplus\!\to Q$ for partially ordered sets (posets), which unifies and generalizes earlier constructions of cobweb posets and their Hasse diagrams (KoDAGs). By enforcing isomorphism conformity on shared sub-posets, the operation enables a simple derivation of the Möbius function formula for cobweb posets and establishes that all graded posets without mute vertices are natural joins of chains of universal relations, yielding explicit formulas for zeta and Möbius matrices via Knuth notation.

ABSTRACT

In addition to the three standard operations on posets which are dual of poset or ordinal and cardinal sums of partial ordered sets one adds the natural join of posets. This is especially natural natural join operation for graded posets including very special and important the so called cobweb posets. The main aims of this article are the presentation of the authors update applications of natural join of posets in the domain of graded posets and summary of the general properties of natural join of posets with posing some questions arising on the way. Thus in this note apart from revealing some general properties of natural join of posets we also deliver the authors combinatorial interpretations of cobweb posets including discrete hyperboxes coding of the so called cobweb posets .Various explicit formulas for the zeta functions and zeta matrices as well as the inverse of zeta matrix for any graded posets are supplied. Whitney numbers formulas and those for characteristic polynomials are given too.

Motivation & Objective

• To formalize and generalize the natural join operation $P \oplus\!\to Q$ for posets, ensuring compatibility via isomorphic sub-posets.
• To provide a unified framework for constructing cobweb posets and their Hasse diagrams (KoDAGs) as natural joins of chains of universal relations.
• To derive explicit formulas for the zeta and Möbius matrices of graded posets, particularly cobweb posets, using Knuth notation.
• To establish that cobweb posets are N-free, series-parallel, greedy, and have dimension two, clarifying their structural and algorithmic properties.
• To pose open problems on tiling, reversibility, and structural properties of KoDAGs and cobweb posets.

Proposed method

• Define the natural join $P \oplus\!\to Q$ such that partial orders on isomorphic sub-posets $R \subseteq P$ and $R' \subseteq Q$ are equivalent, treating $R$ and $R'$ as identical.
• Represent cobweb posets as natural joins of chains of $r$-ary universal relations, where each level $\Phi_k$ has size $k_F$, encoded by a sequence $F = \{k_F\}_{k \geq 0}$.
• Use Knuth notation and upside-down notation ($F_n \equiv n_F$) to express zeta and Möbius matrix formulas for graded posets.
• Apply the natural join to decompose graded posets without mute vertices into chains of binary complete (universal) relations, forming KoDAGs via Cartesian product with projection out of common faces.
• Derive explicit formulas for Whitney numbers and characteristic polynomials of cobweb posets using the natural join structure and matrix supplementation.
• Leverage known results on series-parallel and greedy posets to prove that cobweb posets are N-free and have dimension two.

Experimental results

Research questions

• RQ1 How can the natural join of posets be formally defined to preserve order equivalence on isomorphic sub-posets, and what are its foundational properties?
• RQ2 What is the role of the natural join in simplifying the derivation of the Möbius function formula for cobweb posets?
• RQ3 Are all graded posets without mute vertices representable as natural joins of chains of universal relations, and how does this relate to their Hasse digraph structure?
• RQ4 What is the relationship between the natural join operation and the dimension or Ferrers dimension of a poset, particularly in the case of cobweb posets?
• RQ5 Are cobweb posets reversible, and what conditions determine reversibility in the context of greedy and optimal linear extensions?

Key findings

• The natural join $P \oplus\!\to Q$ satisfies the Natural Join Conformity Condition, ensuring order compatibility on isomorphic sub-posets, and enables a unified construction of cobweb posets.
• Cobweb posets are proven to be N-free and series-parallel, implying they are greedy posets and have dimension two.
• The Möbius function formula for cobweb posets is derived in a simple and direct way using the natural join operation.
• Explicit formulas for the zeta matrix and its inverse (via Möbius function) are provided for any graded poset using Knuth notation and matrix supplementation.
• All graded posets without mute vertices (i.e., no internal vertices with in-degree or out-degree zero) are shown to be natural joins of chains of universal relations.
• Cobweb posets and their Hasse digraphs (KoDAGs) are encoded by discrete hyper-boxes, with the natural join of such hyper-boxes equivalent to their Cartesian product with projection of common faces.